Light control by means of forced translation, rotation, orientation, and deformation of particles using dielectrophoresis

ABSTRACT

Methods and embodiments are provided for the coordinated translation, rotation, and deformation of swarms of nanoparticles by means of forced diffusion by dielectrophoresis in order to affect the scattering of light and the synthesis of the central quantity to all optics: refractive index. Applications include electronic beam steering of light, concentration of sunlight, augmented reality displays, and medical diagnostics, and many others.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims benefit of provisional patent application numberU.S. 62/985,327, entitled “Agile Light Control By Means of OrientationalDielectrophoresis and Electrorotation,” which was filed by Leo D.DiDomenico on 2020 Mar. 5, the entirety of which is incorporated hereinby reference. This application is also a continuation-in-part of U.S.patent application Ser. No. 16/530,889, filed Aug. 2, 2019 and entitled,“Agile Light Con-trol by Means of Noise, Impulse, and Harmonic SignalInduced Dielectrophoresis Plus Other Phoretic Forces to Control OpticalShock Waves, Scattering, and the Refractive Index of Colloids;Applications Include: Solar Electricity, Solar Smelting, SolarDesalination, Augmented-Reality, LiDAR, 3D-Printing, High-Power FiberLasers, Electronic Lenses, Light Beam Steering, Robotic Vision, SensorDrones, Diffraction-Minimizing Light Beams, Power Beaming, andSoftware-Configurable Optics,” which was filed by Leo D. DiDomenico on2019 Aug. 2 and is included herein in its entirety by this reference.

TECHNICAL FIELD OF INVENTION

This disclosure describes the direct electronic control of light bysynthesizing the complex-valued tensor-form of the refractive index inspace and time using nanoparticle distributions, orientations, anddeformable particle shapes in at least one of a gas, a vacuum, asolution, a colloid, and suspension to control scattering and redirectlight and its wavefronts, often along substantially curved trajectories,while also controlling polarization, intensity, loss and gain and otheroptical parameters.

BACKGROUND ART

In U.S. Pat. No. 10,371,936, having patent date 2019 Aug. 6 andentitled, “Wide Angle, Broad-Band, Polarization Independent BeamSteering and Concentration of Wave Energy Utilizing ElectronicallyControlled Soft Matter,” which is included herein in its entirety bythis reference, the current applicant provides a first generation (G1)of a light control technology based on mixing two or more liquidchemicals to form a solution that provides a Refractive Index Matching(RIM) liquid. The G1 technology is characterized by mixing the moleculesof two or more liquid chemicals, both of which have molecules that areless than 1 nm in maximum size. The resulting liquid solution may beadvected through capillaries by various means (e.g. pressuredifferences) to hide and reveal optical boundaries so that totalinternal reflection (TIR) may provide light steering. In subsequentdiscussions G1 may be used to refer to the technology of this paragraphand the patent mentioned herein.

In patent application Ser. No. 16/530,8879, which is described above insection: Cross Reference to Related Applications, the current authorprovides a second generation (G2) of a light control technology based onchanging at least one of the liquid components of the G1 technology to a(typically) solid spherical nanoparticle so that dielectrophoresis (DEP)can provide a means for electronically controlled advection. DEPcomprising conventional DEP that uses two-phase electrical excitation toprovide pondermotive forces in the positive direction of the gradient ofthe square of the electric field (pDEP), or in the negative direction ofthe gradient of the square of the electric field (nDEP). DEP alsocomprising four-phase electrical excitation to provide pondermotiveforces in the direction of a traveling wave (twDEP). Substantial theoryand examples were provided for using nanoparticle spatial translation toaffect control of the scalar refractive index and consequently providelight control. Additionally, the use of electrical noise, which is inthe form of band-limited electric fields, is shown to lower the requiredvoltage levels for DEP in the frequency domain. In subsequentdiscussions G2 may be used to refer to the technology of this paragraphand the patent application mentioned in this paragraph. One of the keydistinguishing features of G1 from G2 is that G1 uses solutions withparticle size less than 1 nm while G2 is based on colloids with particlesize between 1 nm and 1000 nm for visible light optics, but with astrong preference towards particle size of about 30 nm to reduce Tyndalleffect at the same time as increasing DEP forces by a million overparticles in solution.

The reader should note that the current disclosure is about a thirdgeneration (G3) of this author's light control technology. In G3 atleast one of particle translation, rotation, orientation and deformationin an electric field provides a geometric anisotropy and a resultinganisotropy in the refractive index. This allows the use of electricfields to actively take control of the complex-valued tensor form of therefractive index of a colloid by light scattering techniques in such away to electronically manipulate the flow of light directly anddynamically. G2 and G3 in combination allow optics by manipulatingparticles in a colloid. That said, the emphasis in G3 is rotation,orientation, and deformation of particles.

Additional prior art is found in Liquid Crystal Displays (LCDs) andSuspended Particle Devices (SPDs), which are examples of fluidic devicesthat use the asymmetry of a plurality of particles to control light. Thedistinguishing feature of these prior art devices is that they useelectromagnetic fields having zero frequency b=0 electrode excitation.For avoidance of doubt, zero frequency does not mean that there is no ONand OFF switching. Moreover, even if AC fields are used in this priorart, the fields are not used to select the particle orientation as isthe case in the G3 technology of this disclosure where ω≠0.

In practical terms, this means that while LCDs and SPDs have twoorientations the devices of this disclosure can have an infinite numberof orientations. This corresponds to changing the scattering andrefractive index properties continuously and having a continuous controlover light scattering and the effective refractive index of light. Thisincludes the G3 control over optical loss, optical amplification, raytrajectory, wavefront curvature, polarization, and optically appliedforces on particles.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing discussion is only an introduction and other objects,features, aspects, and advantages will become apparent from thefollowing detailed description and drawings of physical principles givenby way of illustration. Note that figures are often drawn for improvedclarity of the underlying physical principles, are not necessarily toscale, and have certain idealizations introduced to show the essence ofthe method and embodiments and to make descriptions clear.

FIG. 1A shows electric fields magnitudes from an array of electrodes.

FIG. 1B shows a nanoparticle concentration as a function of timeresulting from a step input in electric fields for dielectrophoresis.

FIG. 1C shows an example Clausius-Mossotti factor for sphericalparticles.

FIG. 1D shows an example of an optical switch passing light based onparticle translation and light scattering only.

FIG. 1E shows an example of an optical switch reflecting light based onparticle translation and light scattering only.

FIG. 1F shows an example of a structured nanoparticle layer forming agrating that can scatter light.

FIG. 2A shows completely random cylindrical molecules (from a distance)that create an amorphous isotropic material

FIG. 2B shows shows a polycrystalline material that is on averageisotropic.

FIG. 2C shows a crystalline material that is anisotropic except forcubic crystals.

FIG. 2D shows cylindrical molecules that are all oriented along the longaxis and otherwise placed randomly.

FIG. 2E shows cylindrical molecules that are all oriented along theshort axis and otherwise placed randomly.

FIG. 3A shows a finite gap of a first dielectric that is surrounded by asecond dielectric.

FIG. 3B shows an infinite rectangular tube of a first dielectric that issurrounded by a second dielectric.

FIG. 3C shows an infinite cylindrical tube of a first dielectric that issurrounded by second dielectric.

FIG. 3D shows a sphere of a first dielectric that is surrounded bysecond dielectric.

FIG. 3E shows a spheroid of a first dielectric that is surrounded bysecond dielectric.

FIG. 4A shows an ellipse that is used in defining a two dimensionalelliptic-hyperbolic coordinate system.

FIG. 4B shows a hyperbola that is used in defining a two dimensionalelliptic-hyperbolic coordinate system.

FIG. 4C shows an ellipsoid that is used in defining a three dimensionalelliptic-hyperbolic coordinate system.

FIG. 4D shows a hyperboloid of one sheet that is used in defining athree dimensional elliptic-hyperbolic coordinate system.

FIG. 4E shows a hyperboloid of two sheets that is used in defining athree dimensional elliptic-hyperbolic coordinate system.

FIG. 4F shows an ellipsoid, a hyperboloid of one sheet, and ahyperboloid of two sheets together so that the resulting threedimensional coordinate system is shown as the curves of intersection atsingle points.

FIG. 5 shows the depolarization of a spheroid passing from oblate tosphere to prolate. The sphere occurs when r=c/a=1 and M=1/3. The oblatespheroid occurs when 0<G<1 and c<a=b. The prolate spheroid occurs when1<η<∞ and a=b<c. This is critical to forming electric fields inside aparticle that are not in the same direction to the externaldielectrophoresis excitation field to allows non-zero torques onparticles controlling light scattering.

FIG. 6 shows spheroids of the same volume and different shape floatingin a colloid to provide different DEP and optical properties.

FIG. 7A shows a perspective of a plurality of randomly placed prolatespheroids aligned along their long axis.

FIG. 7B shows a side view of randomly placed prolate spheroids alignedalong their long axis.

FIG. 7C shows a side view of randomly placed prolate spheroids alignedalong their short axis.

FIG. 8 shows the trajectories of light in a colloid layer that is usedto properly match the boundary conditions of light propagating initiallyin a homogeneous medium and then in a graded refractive index so as tosteer the light into a desired direction.

FIG. 9 shows the geometry of a colloid gap, which is formed between twotransparent plates, with rays passing through the plates and colloidassuming a positive refractive index gradient and a curved raytrajectory.

FIG. 10 A show in cross section a range of ray behaviors in the presenceof refractive index gradients in a thin colloid layer for the case oflight-beam steering by transmission through the device.

FIG. 10 B show in cross section a range of ray behaviors in the presenceof refractive index gradients and a mirror in a thin colloid layer forthe case of light-beam steering by reflection from the device.

FIG. 11A shows a cross section of light steering panel with randomorientation of nanoparticles in a colloid.

FIG. 11B shows a cross section of light steering panel with electricfields in the colloid control volume to align nanoparticles.

FIG. 11C shows a cross section of light steering panel with alignednanoparticles.

FIG. 11D shows a cross section of light steering panel with perturbationelectric fields sweeping through the colloid control volume.

FIG. 11E shows a cross section of light steering panel withnanoparticles having an angular perturbation.

FIG. 11F shows a cross section of light steering panel with electricfields within the colloid control volume that have at least one ofdifferent field strength, different frequency, and different on-time soas to set the orientation of nanoparticles as a function of position.

FIG. 11G shows a cross section of light steering panel withnanoparticles having different orientations along the colloid controlvolume.

FIG. 11H shows a cross section of light steering panel with analternative angular orientation distribution of nanoparticles along thecolloid control volume.

FIG. 12 shows a cross section of an active colloid lens that focuseslight that passes through its thin layer based in the inducedorientation of nano-particles from dielectrophoresis.

FIG. 13A shows a top view of a reconfigurable waveguide for light.

FIG. 13B shows a cross section view of a reconfigurable waveguide forlight.

FIG. 13C shows a top view of the an electro array of a reconfigurablewaveguide.

FIG. 13D shows a cross section view of a reconfigurable waveguideelectrode array.

FIG. 14 shows a constant gradient in refractive index as implemented inthe orientation of particles and the resulting trajectory of input lightalong an approximately circular trajectory.

FIG. 15 shows a canonical system comprising electrodes, a controlvolume, sources of electric fields, and electric fields of differenttypes for implementing dielectrophoresis.

FIG. 16 shows the same canonical system of FIG. 15, but with an emphasison the particles in the containment volume and the light scatteringprocess.

SUMMARY

Methods and embodiments are provided for the coordinated translation,rotation, and deformation of large numbers of nanoparticles by means offorced diffusion by dielectrophoresis in order to affect the scatteringof light and the synthesis of the central quantity to all optics:refractive index. Applications include electronic beam steering oflight, concentration of sunlight, augmented reality displays, andmedical diagnostics, and many others.

THE WRITTEN DESCRIPTION Foundational Physical Principles Introduction

The technology disclosed herein is named Particle Swarm Optics (PSO) bythis author to reflect its physical presentation of a swarm ofelectronically coordinated particles to affect optics. PSO is used toelectronically synthesize the quantity central to all optics andphotonics: Refractive Index (RI). This is done by means of lightscattering. The objective of the technology is a transparent materialthat is software programmable to become an optical system as itsinternal structure adapts and changes quickly at the scale of opticalwavelengths. This is accomplished by using electronically controlledDielectrophoresis (DEP) to provide forces, torques, and stress onelectrically neutral nanoparticles (NPs) within a colloid or suspensionfor coordinated translation, continuous rotation, transient orientation,and/or deformation of NPs using up to several tens of thousands ofsub-wavelength NPs per cubic micron. Historically, manipulating smallnumbers of cells for medicine, biology, and genetic analysis has beenthe predominant use of dielectrophoresis. However, here DEP is used totypically control “a swarm” of light scattering NPs to synthesizeisotropic and/or anisotropic RI for many optics and electromagneticradiation control applications.

Again, for the avoidance of doubt a solution contains particles that aretypically smaller than 1 nm, a colloid contains particles that arebetween 1 nm and 1,000 nm, and a suspension has particles that arelarger than 1,000 nm. Both the continuous medium and dispersed medium(e.g. particles therein) can be a gas, liquid, and/or solid. Differenttypes of colloids exist. If the dispersed medium is a solid then it iscalled a sol and include: solid aerosol (e.g. smoke, and dust), liquidsol (e.g. paint and milk), solid sol (e.g. ruby glass). If the dispersedmedium is a liquid then the colloids include: liquid aerosols (e.g. fogand mist), emulsions (e.g. milk), and gels (e.g. gelatin). If thedispersed medium is gas then the colloids include: liquid foam (e.g.whipped cream) and solid foams (e.g. aerogel). In this document mostcolloids are liquid sols and emulsions. However, other types of colloidmay in certain circumstances also be applicable. Additionally, largerparticles of suspensions may be used in optical systems under certainconditions. Therefore, when a term like nanoparticle (NP) is used it isto be understood that in certain applications the particle may in factbe larger even many microns in size. Note, for controllingsub-millimeter wave “light” radiation even larger sized particles may beused. Also, for this disclosure the continuous medium may also be avacuum or gas, so that devices can work in space without freezing orboiling.

PSO's historical origin is concentrated solar energy with a firstapplication area of Concentrating Solar Thermal (CST) for intense heatfor industrial electricity, steel smelting, desalination, glass, etc.without fossil fuels. A second application area is display technologysuch as compact virtual reality displays, and RGB laser displays. Athird application area is lasers for cutting, welding, hole drilling,beamed energy-distribution networks, high-energy fiber bundle laserswith cladding-based RI phase shifters & thermal compensation, activebeam-combiners, solid/wet-state beam directors, self-healing optics, andwavefront compensators. This impacts areas such as beamed energydistributions networks for powering airborne drones. A fourthapplication area is LiDAR transceivers, THz-Wave beam steering,broad-band optical antennas, inertial sensors, single-pixel cameras. Afifth application area is information technologies such as opticalneural networks, quantum computing optics, 5G Light Fidelity (LiFi)networks. A sixth application area is meta-materials such for adaptivethermal signature control via fluidic photonic band-gap meta-materials.A seventh application area is optics for very large apertureadjustable-membrane-optics for satellites & astronomy, dynamic free-formoptics, glancing-angle reflections for y-ray astronomy, adaptivelighting, negative RI optics induced by by resonant NP clusters, anddynamic holograms. Many other features and applications exist.

PSO is based on DEP, which provides forces, torques, and stresses onuncharged particles. The forces are on electrically neutral particlesand in the direction of the electric field's energy gradient and not inthe direction of the electric field itself. For the forces to be in thedirection of the field itself the particles would have to be charged,and this would imply electrophoresis (nota bene!!) not dielectrophoresisas the governing process. Furthermore, torques require anisotropic mediaand/or non-spherical NP shapes so the electric field internal to theparticle is not in the same direction as the electric field external tothe NP.

The simplest PSO system is based on a paper-thin colloid sandwichedbetween two transparent glass sheets with transparent electrodes printedon the glass. Light passing through the panels is redirected, focused,and/or wavefront compensated depending on the spatial distribution ofNPs and their orientation. Coordinating this is a microprocessor thatconverts the desired behavior of light into a RI distribution in spaceand time. This is then converted into an array of oscillating voltageamplitudes, phases, and frequencies that are applied to transparentpixelated electrodes to create the needed electric fields to control DEPforces F and torques T on the NPs in the colloid. The resulting forcesand torques on NPs in the liquid causes changes on at least one of aNP's position, orientation and shape so that light scattering phenomenaprovide an effective average RI of the colloid from point-to-point toprovide a controllable graded refractive index. For example, a colloid'sliquid component may have a low RI of about 1.4, and its NP componentshave a high RI of almost 2.9 so that the range of the synthesized RI is1.4≤RI≤2.4, though multi-scattering and other effects may decrease thisrange somewhat. Simple applications using dielectric NPs includesteering, focusing, and general wavefront control of light. The use ofmetals, dielectrics, semiconductors, crystals like titanium dioxide,cage molecules (e.g. Bucky balls), cluster-particles with resonances,electrets, biologically self-assembled particles, quantum dots,plasmonic dots, doped glass with optical gain, bubbles, viruses,polyimide NPs, and other NPs and NP combinations can extend opticalfunctionality into advanced areas such as laser sources, neuralnetworks, and quantum computing.

Stable colloids with maximum NP size, typically between 30 nmnmdiameter, are used for visible light RI synthesis and should ideallystay in the colloidal suspension indefinitely. The colloid must also beoptically clear and of low optical insertion loss at 10 μm colloidthickness and have the correct conductivity and permittivity of liquidand NPs to create the correct response of the electric susceptibilityfor DEP force control by impressing different frequencies. Colloids mustnot aggregate via Van der Waals, Casimir, and other short range forces;do not present an undesired color due to light-absorbingcrystal-structure defects from NP manufacture, and have a number ofphysiochemical properties related to fluids, particle size, shape,composition, and non-toxicity. Another set of complex considerationsalso exist for the choice of the liquid. Some of the issues includeviscosity, temperature stability, chemical stability, photo-stability,high transmissivity, ionic vs. covalent liquids, Newtonian fluidschanging to non-Newtonian fluids at the nano-scale, toxicity, andothers. An example of this colloid synthesis complexity is the use ofirradiation of newly fabricated NPs by intense y-rays, X-ray,ultraviolet light, and/or neutron bombardment to reconfigure crystalstructure and convert dark-colored absorptive colloids into highlytransparent optical materials. High-quality colloids have beendemonstrated in this way in academic and industrial labs. Also, theminimization of undesired Tyndall scattering by a careful selection ofparticle size and its bulk RI is an important consideration.

Additionally, DEP excitation frequencies typically occur between 1 kHzand 1 GHz at 1 KV. However, a noise voltage can trade kV-level sinewaves for logic-level signal sources using an easily generated largebandwidth. For example, a 1 kV sinusoidal signal can be converted into amodest 1 V/√{square root over (Hz)} noise level at 1 MHz bandwidth thisallows low-cost electronics.

Next, a survey of equations and processes to synthesize RI based on NPdiffusion of spheres, prolate spheroids, and oblate spheroids follows.In particular, the state of colloid can be represented by two or moreprobability density functions (PDFs). The first PDF ƒ_(F)×F_(F)(r, t, ω)is for NP positions and connects to particle translation by DEP inducedforces. The second PDF ƒ_(T)=ƒ_(T)(r, t, θ, ϕ, ω) is for NP orientationand connects to particle rotation by DEP induced torques. In theseequations r is position of a differential volume within the colloid, tis time, a NP's zenith angle is θ, a NP's azimuth angle is ϕ, and c isthe DEP electric field excitation frequency.

Further, Re[·] is the real part function, F is the average force on aNP, T is the average torque on a NP, ∈_(L)(ω) is the permittivity of theliquid at DEP frequency b, V is the volume of a NP, χ(ω) is the NP'stensor electric susceptibility at DEP frequency ω (e.g. typically up to1 kHz to 1 GHz) due to NP shape, E_(in)(r) is the input electric fieldphasor at DEP frequency, up is the volume fraction of the NPs, ξ is theaverage rotational diffusion tensor, W is a rotation matrix in sphericalcoordinates, γ_(F) is a NP's linear mobility and γ_(T) is the NP'srotational mobility, n _(A) is the average complex tensor form of the RIof the colloid, n_(L) is the liquid's scalar RI, I is the identitymatrix, n_(p) is a NP's effective RI including depolarization, and λ theoptical wavelength. Additionally, gradients ∇_(X) and ∇_(Θ) usecartesian and spherical coordinates respectively.

With those definitions made it can be shown that DEP forces require anenergy gradient proportional to ∇_(X)|E_(in)|² inside NPs whereby

F=Re[∈_(L)ξχ∇_(X) |E _(in)|²]/2  (1)

(see Eq. 306). However, DEP torques require a field-direction changeinside of each NP (i.e here is the field direction change is χE_(in)),and it is later shown that

T=Re[∈_(L) νLV(χ E _(in))×E _(in)]*/2  (2)

(see Eq. 312). Moreover, the electric susceptibility has two special(extreme) cases in the symmetry coordinates of the NPs

$\begin{matrix}{\overset{=}{\chi} = \{ \begin{matrix}{N\; \overset{=}{\alpha}} & {{Low}\mspace{14mu} {NP}\mspace{14mu} {volume}\mspace{14mu} {fraction}} \\{( {N\; \overset{=}{\alpha}} )\lbrack {\overset{=}{I} - {\overset{=}{M}( {N\; \overset{=}{\alpha}} )}} \rbrack}^{- 1} & {{High}\mspace{14mu} {NP}\mspace{14mu} {volume}{\mspace{11mu} \;}{fraction}}\end{matrix} } & (3)\end{matrix}$

where a is a NP's polarizability, N is the NP concentration, M is ageometric factor called the depolarization factor that is based on theshape of NPs. These two regimes can provide significantly differentkinds of electrokinetics: with and without torques respectively.

Additionally, sums of forces and torques using different DEP frequenciesmay lower the required source voltages and/or current sources in somecases so that

$\begin{matrix}{F_{tot} = {{\sum\limits_{j}{{F( {r,\omega_{j}} )}\mspace{14mu} {and}\mspace{14mu} T_{tot}}} = {\sum\limits_{j}{{T( {r,\omega_{j}} )}.}}}} & (4)\end{matrix}$

For example, forces are proportional to the voltage squared, thereforethree sinusoidal voltages sources each of V₀/√{square root over (3)} anddifferent frequencies {ω₁, ω₂, ω₃} can provide the same force/torque ona NP as a single voltage source V₀ ² at ω₀ because(V₀/3)²+(V₀/3)²+(V₀/3)²=y, which is the original excitation.

Translation diffusion then links total force F_(tot) to its PDF

∂_(t)ƒ_(F) −D _(F)∇_(X) ²ƒ_(F)=−∇_(X)·[γ_(F)ƒ_(F) F _(tot)]  (5)

(see Eq. 378) and orientation diffusion links total torque T_(tot) itsPDF

∂_(t)ƒ_(T) −D _(T)∇_(Θ) ²ƒ_(T)=−∇₇₃·[γ_(T)ƒ_(T) T _(tot)].  (6)

(see Eq. 379). The NP volume fraction v_(P) is then given in terms of aforce PDF and a uniform background volume fraction v₀:

v _(P)(r,t,ω _(j))=v ₀ƒ_(F)(r,t,ω _(j))  (7)

Additionally, an average NP rotation tensor ξ is given by averagingrotations using the torque PDF

ξ(r,t,ω ₀)=∫∫{dot over (W)}(θ,ϕ)ƒ_(T)(r,t,θ,ϕ,ω _(j))dθdϕ.  (8)

Finally, a RI Tensor n _(A)(r, t, ω_(j), ξ) is given in terms of v_(P)and {dot over (ξ)}

n _(A) ≈n _(L) I +[ n _(P)(ξ)−n _(L) İ]σ_(P)(ξ)v _(P)  (9)

where σ _(P)(ξ) is the tensor shape fraction that depends on thegeometric profile of a NP as seen from different directions—see Eqs.270-273. Note that the two quantities that are controllable byelectronics are v_(P) and ξ, for translation and orientationrespectively. Thus, DEP forces and torques (Eqs. 1, 2, & 4) on NPs causethe colloid's state to change via two or more Fokker-Planck diffusionequations (Eqs. 5 & 6) that connect volume fraction v_(P) and a particleorientations ξ (Eqs. 7 & 8) to the RI (Eq.) and diffusion.

Note that the NPs need not be solid. For example a class of colloidsthat are soft are emulsions. When a particle forming an emulsion issubjected to DEP forces it may deform so that NP shape is also subjectto change and described by the tensor shape fraction σ _(P). Therefore,another diffusion equation forced by the Maxwell stress tensor controlsanother diffusion process.

In general the average RI tensor n _(A) provides the information neededfor programming light propagation by a scattering process e.g. evenbending light. Typically, the liquid has a scalar RI n_(L), but thisalso need not be the case in general wherein n _(L) may be used. Thus,by this survey of PSO equations, forced NP diffusion for NP translation,rotation, orientation, deformation, and combinations thereof can controlrefractive index which is the single most important quantity for opticsand the control of light.

For the avoidance of doubt, “light” means visible and non-visible formsof electromagnetic radiation, such as: x-rays, ultraviolet, infrared,visible from about 380 nm to 700 nm, sub-millimeter wave (sometimescalled T-wave) and other bands of the electromagnetic radiationspectrum. Translational and orientational DEP are closely related andoften are configured to work together to produce the desired RI andlight scattering.

Examples of RI synthesis using spherical NPs and translation only areshown in FIG. 1, where DEP forces on 40 nm diameter NPs energized by 1kV sine wave or equivalently a 1 V/√{square root over (Hz)} noise at 1MHz bandwidth, are by order of magnitude: 100 million times greater thangravity forces, 1 million times greater than the DEP forces on liquidmolecules surrounding NPs, and 1000 times greater than thermal forces at300 K near electrodes having a 10 μm period and 50% duty cycle; withforces decaying to about 100 times greater then thermal at the otherside of the colloid sheet 10 μm away.

FIG. 1A shows a cross sectional view of a 10 μm thick colloid layer withelectrodes at y=0 (black rectangles) and iso-contours of the magnitudeof the electric field. DEP forces are in the y-direction because thegradient of the electric field magnitude is substantially in they-direction and normal to the iso-contours. FIG. 1B shows solutions ofthe translational Fokker-Planck equation in time and in response to aunit step in a harmonic electric field using 0.1 kV amplitude sinusoidalexcitation without the advantage of using a noise signal to reducevoltage amplitude to about 1 V/Hz. Initially the NP volume fraction is10%, corresponding to the FIG. 1D NP distribution. Then after a periodof time the colloid settles to a steady-state where about half thecolloid region is near 0% volume fraction and the other half is near 20%volume fraction, corresponding to the NP distribution in FIG. 1E. Next,FIG. 1C shows the normalized real and imaginary parts of the electricsusceptibility χ(ω) for a spherical NP colloid. Historically these arethe Clausius-Mossotti factors {K_(R), K_(I)} and ω₀=10 MHz aredetermined by the permittivity and conductivity of the NPs and liquidcomprising the colloid. By choosing the excitation frequency at ω₀/10(point a) NPs are pushed away from the electrodes and by choosing theDEP excitation frequency at 10ω₀ (point b) the NPs are pulled towardsthe electrodes. This can be used to make an optical switch. FIG. 1Dshows the colloid layer with NPs dispersed uniformly throughout. Lightpasses through without deviation because the RI of the colloid (1.55),i.e. n_(A)=n_(L) (n_(P)−n_(L))v_(P)=1.40+(2.9−1.4)(0.1)=1.55, is matchedto the surrounding glass (1.55). This corresponds to the initialconditions of FIG. 1B. In FIG. 1E a sinusoidal excitation at ω=10ω₀ withalternating polarity between electrodes excites the colloid to retractNPs from the boundary of the glass and colloid. This leaves a RI of 1.40in the colloid at the optical boundary with RI of 1.55 so that TotalInternal Reflection (TIR) reflects the incident ray. In this particularexample the maximum NP speed, when using 100 V amplitude voltages, isestimated to be about 1 μm·ms⁻¹. Therefore, by switching the DEPfrequency between ω₀/10 and 10ω₀, an optical switch is formed, with anestimated switching speed of about 1 ms. Alternately, at 1 kV amplitude(this can be reduced to a modest 1 V/√{square root over (Hz)} noisesignal at 1 MHz noise bandwidth) NP velocity is estimated to be about100 μm·ms⁻¹, with optical switching speeds on the order of 10 ns. Higheramplitude fields and/or larger voltage bandwidths can improve this speedsignificantly. Finally, in FIG. 1F an electrically large(non-diffractive) light-steering grating is shown as an example of whatmay be possible with more electrodes to better generate a spatiallyperiodic grating by Fourier superposition of fields and by controllingfrequencies, magnitudes, and phases to synthesize a desired3-dimensional RI distribution. Translational DEP RI synthesis are alsopossible using torques and orientation of NPs and this is the topic ofthe remainder of the majority of this disclosure.

The Scope of Dielectrophoresis

The name dielectrophoresis (DEP) is composed of three morphemes di-,which comes from the latin and means “apart from;”-electro-, which comesof the Greek word amber and relates to phenomena associated withelectrons that may be rubbed off thereof by the triboelectric effect;and -phoresis, which comes from the Greek language and means “to carryaround.” Thus, dielectrophoresis means that which is apart from a singleelectron or charge and which provides a means to move or carry around.It should be noted that the morpheme di- can also come from the Greek,and when it does it means a quantity of two, which is cognate with theprefix bi-. Finally, the prefix di- also has a second meaning in theGreek, which is “through and in different directions.” Thus, DEP is aphenomena that does not apply to materials with a built-in single charge(like ions), but which does allow pairs of induced charges (i.e. anelectric dipole moment) to assist in carrying and manipulating matter (aparticle) through other media e.g. a liquid, gas, etc.

The meaning of dielectrophoresis continues to evolve. Variations ofdielectrophoresis include: standing wave dielectrophoresis (sDEP),traveling wave dielectrophoresis (tDEP or twDEP), negativedielectrophoresis (nDEP) where particles translate opposite to thegradient of the electric field, positive dielectrophoresis (pDEP) whereparticles translate in the direction of the gradient of the electricfield, rotational dielectrophoresis (rDEP), electro rotation (ROT orEROT), orientational dielectrophoresis (oDEP), and others.

This proliferation of terms over time is an indication of a healthy andgrowing field where new insights are constantly developing. Thus, theword “dielectrophoresis” from the 1970's does not hold the same nuancedmeaning as “dielectrophoresis” in the 2020's. Originally,dielectrophoresis was developed by Herb Pohl and his graduate studentsat Oklahoma State University (USA) around 1966 and was originallyassociated with translation of a particle (e.g. a blood cell) due to anonuniform electric field. However, now around 2020 it can also includerotation, orientation, and deformation of a particle. With so manypotential types of dielectrophoresis it often makes sense to talk abouta generalized dielectrophoresis (gDEP) that includes all possiblephenomena. When the specific form of dielectrophoresis is not obviousfrom the context of the writing then the casual term dielectrophoresis(DEP) will mean gDEP in this document. The reader is cautioned thatdifferent authors may introduce their own notation for the differenttypes of gDEP modes of operation.

System of Units

For the avoidance of doubt all equations herein, including Maxwell'sequations and electromagnetic quantities in general, are provided in theinternational system of units (SI) unless stated otherwise.

Phasor Math

For the avoidance of doubt phasor representations of mathematicalquantities always assume that ω is the radian frequency and t is thetime so that the time phasor factor is e^(iωt), where i=√{square rootover (−1)} is the physicist's typical convention so that the use of theletter j (often used by engineers for the complex unit) is reserved forelectric and other types of particle currents.

Operators

For the avoidance of doubt the multiplication of a matrix A by a vectorx may be written either as A·x or as Ax, because at times onerepresentation may be more intuitive than the other. Note that whilex·y=y·x, it is also true that A·x≠x·A. Many other combinations of tensorproducts are also possible and the use (or lack of use) of an explicitdot product in the equations will also be based on context and thisauthor's subjective sense of what's needed for clarity.

Time, Frequency, and Spatial-Frequency Domains

For the avoidance of doubt, note that the time domain is represented byscript variables and the frequency (phasor) domain is represented byroman characters. For example, the electric field in the time domain isξ(r, t) and in the frequency domain it is given by ξ(r, c). In thosecases where the spatial frequency domain is also needed it can bedetermined from context. Thus if the spatial frequency domain is alsoused then the electric field is written as ξ(k, t), where k is thevector wavenumber. The one exception to this rule is a particle's volumeV.

At Least Two Frequency Bands

For the avoidance of doubt, for optical applications there are twotemporal frequency bands: the Dielectrophoresis (DEP) frequencies thatare often (but not necessarily) in the range of 1 KHz to 1 GHz and theoptical frequency 430 THz-750 THz (terrahertz) controlled by NPs. Theprimary characteristic is that these frequency domains are so far apartthat a material's constitutive parameters are often much different andset by different material resonances and/or oscillation modes. Todistinguish the two domains the symbol ω is used to describe a DEPexcitation frequency and symbol λ is used to describe the optical wavelength. For example, the average RI of a DEP controlled colloid is givenby n_(A)=n_(A) (r, t, v_(S), λ, ω, T). Even in the Terahertz-wave(T-Wave, millimeter-wave) region where applications utilize microparticles instead of nanoparticles, there is still a large differencebetween the DEP electrokinetic frequency and the control of the opticalor T-Wave energy.

At Least Three Coordinate Systems Per Particle

For the avoidance of doubt, note that there are at least threecoordinate systems for a particle. First is the observer coordinatesystem. Second is the geometric eigen symmetry basis of thenanoparticle. Third is the eigen symmetry basis of the crystallographicsystem comprising the material in the particle. Additional coordinatesystems are needed for each additional layer or other geometric shape orcrystal added. Therefore, the orientation of a crystal structure, aparticle geometry, and the observer can be specified independently. Forexample, a uniaxial crystal in a prolate spheroidal particle can bealigned so that the index ellipsoid of the RI of the crystal is alignedor perpendicular to the major axis of the prolate spheroid to inducespecific optical scattering properties. The observer system has noprimes, e.g. A _(R). The symmetry basis of the particle has one prime torepresent it, e.g. A _(R)′. The symmetry basis of the crystalarrangement making up the material in the particle has two primes torepresent it, e.g. A _(R)″. Primes are not used in this document torepresent real and imaginary parts.

Real and Imaginary Parts of a Quantity

The real and imaginary parts of a quantity are represented by asubscript of R and I respectively. For example the real and imaginaryparts of the tensor Clausius-Mossotti factor in the eigen symmetry basisof a particle are K _(R)′ and K _(I)′ respectively, where the doublebars above a variable indicate tensor quantity.

Electromagnetic Scale

For the avoidance of doubt, Maxwell's equations are modified to ensurethat the speed of light between particles is not necessarily taken asthe speed of light in vacuum, as is done conventionally. Rather, it isassumed that if a discussion involves a liquid between NPs then the“background” speed of light is that in the liquid not the vacuum. Thiswill save countless pages of mathematical analysis and make theunderlying physics much more understandable. Thus, if D is the electricdisplacement, E the electric field intensity, P the polarization densityand ∈₀ is the permittivity of free-space, then instead of writingD=∈₀E+P we write D=∈_(med)E+P, where ∈₀→∈_(r)≡∈_(med), and ∈_(r) isdielectric constant of the medium outside a particle. We can set thevalue of ∈_(med) as needed to make our work efficient. This may be for avalue consistent for vacuum ∈₀ or something else, such as the liquidsurrounding a NP wherein we would set the medium permittivity to theliquid permittivity ∈_(med)=∈_(L). If no choice has yet been made thenthe generic value of ∈_(med) is used. More will be said later about thistopic as needed. In summary, PSO covers many orders of physicalmagnitude. From atomic scale to kilometer scale for applications. Thus,it is practical to only worry about vacuum light propagation only whenneeded. For example, it is quite irrelevant that there is a vacuumbetween molecules in a liquid if one is only interested in macroscopiclight-matter interactions in a liquid. At other times the vacuum betweenthe molecules might be important, e.g. RI derived from thepolarizability.

Constitutive Equations in the Temporal Frequency Domain

Maxwell's equations in SI units are

$\begin{matrix}{{\nabla{\times {ɛ( {r,t} )}}} = {- \frac{\partial ( {r,t} )}{\partial t}}} & (10) \\{{\nabla{\times ( {r,t} )}} = {{+ \frac{\partial ( {r,t} )}{\partial t}} + {_{f}( {r,t} )}}} & (11) \\{{{\nabla \cdot}( {r,t} )} = {\rho_{f}( {r,t} )}} & (12) \\{{{\nabla \cdot}( {r,t} )} = 0} & (13)\end{matrix}$

and the constitutive relations are

(r,t)=∈(r,t)ξ(r,t)  (14)

(r,t)=μ(r,t)

(r,t)  (15)

(r,t)=σ(r,t)ξ(r,t)  (16)

The electromagnetic quantities above include the electric fieldintensity ξ [V/m], the electric field density

[C/m²], the magnetic field intensity

[A/m], the magnetic field density

[Wb/m²], the free electric current density

_(ƒ) [A/m²], the free electric charge density ρ_(ƒ) [C/m³], thepermittivity ∈ [F/m], the permeability μ [H/m] and the conductivity σ[S/m]. If we assume that quantities are harmonic then we can decomposethe space and time functions of fields and charges into functionalproducts in phasor space and material properties into functions of spaceand frequency, which are to be calculated or assumed as the situationrequires, so that in the frequency domain we have

(r,t)→D(r)e ^(iωt)  (17)

ξ(r,t)→E(r)e ^(iωt)  (18)

(r,t)→B(r)e ^(iωt)  (19)

(r,t)→H(r)e ^(iωt)  (20)

ρ_(ƒ)(r,t)→ρ_(ƒ)(r)e ^(iωt)  (21)

σ(r,t)→σ(r,ω)  (22)

∈(r,t)→∈(r,ω)  (23)

μ(r,t)→μ(r,ω)  (24)

where quantities like ξ(r) are in general complex-valued quantitiesunless noted otherwise and i=√{square root over (−1)}. Alternately, wecan write that ξ(r, t)=Re [E(r)e^(iωt)].Either way we can rewrite Maxwell's equations as

∇×E(r)=−iω{tilde over (μ)}(r,ω)H(r)  (25)

∇×H(r)=−iω{tilde over (∈)}(r,ω)E(r)  (26)

∇·{{tilde over (∈)}(r,ω)E(r)}=0  (27)

∇·{{tilde over (μ)}(r,ω)H(r)}=0  (28)

where the third and fourth equations above follow from taking thedivergence of the first and second equations and using the vectoridentity ∇·(∇×A)=0, where A is an arbitrary vector over spacecoordinates. Furthermore, we find that on using the constitutiverelation

_(ƒ)=σ_(ƒ)ξ that Eq. 11 takes the form of Eq. 26 when

$\begin{matrix}{{\overset{\sim}{\epsilon}( {r,\omega} )} = {{\epsilon ( {r,\omega} )} - {i\frac{\sigma_{f}( {r,\omega} )}{\omega}}}} & (29) \\{= \ {\{ {{\epsilon_{R}( {r,\omega} )} - {i\; {\epsilon_{I}( {r,\omega} )}}} \} - {i\frac{\sigma_{f}( {r,\omega} )}{\omega}}}} & (30) \\{= \ {{\epsilon_{R}( {r,\omega} )} - {i\{ {{\epsilon_{I}( {r,\omega} )} + \frac{\sigma_{f}( {r,\omega} )}{\omega}} \}}}} & (31)\end{matrix}$

The above equations are provided to establish notation and signconventions such as the phasor sign convention given by e^(iωt) and toexplicitly show how the effective permittivity is a function offrequency b.

Point Charges

Consider an electrically neutral NP adrift in a liquid and subject to anon-uniform input electric field ξ_(in)(r, t) at a point r and at timet. The NP responds to the electric field in the fluid medium andreconfigures its bound electric charges so that the NP becomespolarized, while still remaining neutral. There exists an average pointcharge that can be used to represent the charge distribution. Thischarge is the spatial average of the positive (or negative) charges sothat

$\begin{matrix}{{q(t)} = {\frac{1}{\nu}{\int_{NP}{{\rho ( {r,t} )}d^{3}r}}}} & (32)\end{matrix}$

where r is the position of the differential elements of the chargedistribution, t is time, q(t) is the magnitude of the time varying pointcharge, ν is the volume of the NP, and ρ(r, t) is the time varyingmagnitude of the positive (or negative) spatial charge density in unitsof charge per unit volume, which is derived from an exact solution ofMaxwell's equations.

Polarizability and the Central Theme

PSO utilizes electromagnetic (EM) energy of one form to control EMenergy of another form through suspended particles. For example, theparticles may be nanoparticles (NPs) or microparticles (MPs). Thesuspending medium may be a liquid, a gas, vacuum, magnetic fields, orother. The EM radiation may be low frequency RF energy or high frequencyoptical energy. The particles are controlled in position, orientation,and deformation by EM fields to subsequently control optical and otherforms of EM radiation through a controlled and coordinated lightscattering process. For a liquid medium with NPs therein we create adynamic Fluidic Meta Material (FMM).

Furthermore, as will be derived later in this document, there are tworelations that are at the heart of manipulating an optical particle byforces and torques. In particular, the force F and torque T are given by

F=p _(par) ·∇E _(in)  (33)

T=p _(par) ×E _(in)  (34)

where E_(in) is the input electric field and p_(par) is the resultinginduced dipole moment of the particle. Thus, much of this document isabout calculating p_(ind) for different types of optical particles undervarious conditions and then coordinating the resulting forces andtorques to create a desired optical effect.

Different Types of Materials

The macroscopic phenomena of RI, and generalized DEP work together tomake PSO functional. These processes are a result of the tensorpolarizability a of materials, which links the atomic, molecular, and NPscales to the macroscopic scale. Thus, the polarizability α is centralto understanding and designing PSO based systems.

The polarizability of a material is a result of several processesincluding electron cloud displacements relative to atomic nuclei, atomicdisplacements relative to other atoms, and permanent built-in dipolemoments in molecules due to chemical bonding from unbalanced chemicalelectronegativity. Polarizability α is also a result of the shape ofNPs, which is expressed by a depolarization factor, and the direction ofthe input electric field. These factors influence the spatialdistribution of surface charges surrounding each NP. Furthercomplicating a material's net polarizability are various types of atomicand molecular long-range interactions forming: pure crystals,polycrystallines, mixtures of ordered crystals and amorphous materials,and pure amorphous materials. See FIG. 2, which shows several differenttypes of long range order in a material. (A) shows completely randomcylindrical molecules (from a distance) that create an amorphousisotropic material, (B) shows a polycrystalline material that, onaverage, is isotropic, (C) shows a crystalline material that isanisotropic (except for cubic crystals), (D) shows cylindrical moleculesthat are all oriented along the long axis in the y-direction andotherwise placed randomly, and (E) shows cylindrical molecules that areall oriented along the short axis in the y-direction and otherwiseplaced randomly. Given the many details of a complex-structured colloid,a unifying model for PSO brings together the many disparate ideas andwill give the reader a greater understanding, which is needed to learnthis topic. This material is discussed in detail shortly and thenconnected to light scattering, RI, DEP, and ultimately to devices andhardware.

Active and Passive Transformations of Particles and Fields

The forces, torques, and stresses on a NP are described in terms of thelocal fields of the eigen symmetry basis of each NP. Thus, a passivecoordinate transformation is needed to map the applied electric fieldsin the observer's reference frame into each particle's eigen symmetrybasis without the particles physically moving. Afterwards, the changingpositions, orientations, and shapes of NPs are described in theobserver's coordinate system using an active coordinate transformationof each NP in response to the locally applied fields. Thus, both passiveand active coordinate transformations are in principle needed repeatedlyto show how a system evolves in time.

Obviously, it is not practical to track the individual positions,orientations, and shapes of potentially trillions of particles in acolloid. Therefore, probability distribution functions are often used totrack averages and standard deviations from the average in eachdifferential volume element of the colloid. For example, if theorientation of NPs in a differential volume element of the colloid areuniformly distributed over 4π steradians then any NP orientation is agood starting place for calculations. Alternately, if the averageorientation of NPs in a differential volume element of the colloid ismostly in a specific direction then, that orientation can be used as thestarting point of analysis. Therefore, in this document the NPs areassumed to have known probability distribution functions of position,orientation, and shape at time t=0, to overcome the difficulty ofseparately setting the state of so many separate NPs. Any errors thisincurs during the time the colloid is changing state is not importantbecause it is usually the case that the steady-state probabilitydistributions are what is important in optics and this will be well knowfrom the stochastic analysis.

Cavity Electric Fields

When considering the electromagnetic fields within a medium it istypical to make approximations to avoid all the details of the actualfields. Otherwise, the complexity of all the fields between atoms andmolecules would become overwhelming. For example, consider a dielectricbetween two conducting capacitor plates separated by a distance d andenergized with a voltage difference V, an average electric field isoften taken as V/d. However, this zero order approximation goes too far,i.e. it is too simplistic, when the individual behavior of particles isdesired even in a simple uniform electric field. This is because eachparticle is also influenced by the fields of its neighbors, and thisincludes the shapes of particles. This is especially important in adense material comprising solids and liquids.

To gain access to the electric fields of the neighboring particles,first consider a neutral medium that has no built-in dipole moments,e.g. a beaker of silicone oil. First, impose an input electric field∈_(in) the medium to induce dipoles within the medium. Second, a smallvolume of the medium is mathematically removed by “freezing” all dipolesand removing particles to form a small cavity. It should be clear that

E _(in) =E _(par) +E _(cav)  (35)

where the fields from the particles removed is E_(par) and the fieldsfrom the cavity formed is E_(cav). To extend this line of reasoning to amixture such as a colloid, comprising a solid particle and surroundingliquid we imagine a third step of replacing the resulting vacuum cavitywith the original liquid (less any induced dipoles). Then the cavityfields are in a cavity with a liquid background.

The electric fields of E_(cav) and its relation to the shape of theparticle is now developed gradually by considering some simple examplescomprising different cavity shapes as shown in FIG. 3 and calculatingthe cavity electric field E_(cav) assuming only induced dipoles andbackfilling the cavity with the medium surrounding the cavity. Note thatin the examples associated with FIG. 3 it is assumed that the eigensymmetry basis (coordinates) of a particle is the same as the observerscoordinates so that x=x′, y=y′, and z=z′, this will simplify notationinitially.

Example A corresponds to FIG. 3A, which shows a thin cavity between twodielectrics. The cavity is parallel to the xz-plane and its thickness isin the y-direction, where the coordinates are the particle's symmetrycoordinates. For a z-directed input electric fieldE_(in)=E_(in){circumflex over (z)}, where {circumflex over (z)} is az-directed unit vector we can apply the integral form of Stokes' lawϕ_(C)E_(in)·dl=0 along a contour, which is shown in FIG. 3A. The contouris parallel to the yz-plane with one part of the contour inside thecavity and one part of the contour outside the cavity. Therefore,E_(in)·{circumflex over (z)} dl+E_(cav)·(−{circumflex over (z)}) dl=0 sothat E_(cav)=E_(in). Which we can write asE_(cav)=E_(in)−(0)P_(ind)/∈_(med) to highlight its relation to thepolarization density. Therefore, in this particle's symmetry basis wefind that M_(xx)=0. Also, by symmetry we find that M_(zz)=0. To find theinduced electric field in the y-direction Gauss's law may be used. Inparticular, by Gauss's law ∫_(A) E_(in)·dA=Q_(tot), where A/2 is thearea of each gap surface within the bounding box, and Q is the magnitudeof the charge on each surface. Therefore, σ ₁=Q/(2A) on one surface andσ₂=−Q/(2A) on the other surface. The magnitude of the total polarizationdensity in the cavity is (Q/A), whereby P_(cav)=−(Q/A)ŷ is in thedirection from negative to positive induced charges and the inducedelectric field in the cavity is (−P_(cav)/∈_(med)). Thus, if the inputfield in the cavity is in the y-direction thenE_(cav)=E_(in)−(1)P_(cav)/∈_(med). In anticipation of generalizing thisresult we can write

$\begin{matrix}{E_{cav} = {E_{in} - {\frac{1}{\epsilon_{med}}\underset{\underset{\overset{=}{M}}{}}{\begin{bmatrix}0 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 0\end{bmatrix}}{\underset{\underset{P_{cav}}{}}{\begin{bmatrix}P_{{cav},x} \\P_{{cav},y} \\P_{{cav},z}\end{bmatrix}}.}}}} & (36)\end{matrix}$

Where M is the tensor depolarization that is given in the symmetry basisof the cavity. In most calculations this will be transformed to a moreappropriate coordinate system as explained later.

Example B corresponds to FIG. 3B, and is very similar to the priorexample, except that now the total surface area within the bounding boxis 2(A_(x)+A_(y)) so that

$\begin{matrix}{E_{cav} = {E_{in} - {\frac{1}{\epsilon_{med}}\underset{\underset{\overset{=}{M}}{}}{\begin{bmatrix}{A_{x}/( {A_{x} + A_{y}} )} & 0 & 0 \\0 & {A_{y}/( {A_{x} + A_{y}} )} & 0 \\0 & 0 & 0\end{bmatrix}}{\underset{\underset{P_{cav}}{}}{\begin{bmatrix}P_{{cav},x} \\P_{{cav},y} \\P_{{cav},z}\end{bmatrix}}.}}}} & (37)\end{matrix}$

Example C corresponds to FIG. 3C for an infinite cylinder. In the zdirection the analysis is very similar to that in Example A where Stokeslaw along a contour is used and we find that a z-directed input field isequal to the cavity field. However, an input field in either the x or ydirections must provide the same local field inside the cylinder. Forexample, if E_(in)=E_(in){circumflex over (x)} then induced polarizationcharges on the cavity are positive for x>0 and negative for x<0, asshown in the figure. Let ϕ be the azimuth angle measured from thepositive x-axis towards the positive y-axis, then the distance betweeneach charge is d=d₀ cos ϕ and the charge is q(ϕ)=q₀ cos ϕ; where d₀ isthe diameter and q₀ is the maximum charge separated. Therefore, thedipole moment is p=q d{circumflex over (x)}=(p₀{circumflex over (x)})cos² ϕ. Therefore, the average dipole moment is

$\begin{matrix}{{\langle p\rangle} = {{\frac{1}{\pi}{\int_{\pi/2}^{3{\pi/2}}{p_{0}\cos^{2}\varphi d\varphi}}} = {\frac{( {p_{0}\hat{x}} )}{2}.}}} & (38)\end{matrix}$

so that P=N(p₀{circumflex over (x)})/2 and the induced electric field isP/∈_(med). Similar equations exist for the y, therefore

$\begin{matrix}{E_{cav} = {E_{in} - {\frac{1}{\epsilon_{med}}\underset{\underset{\overset{=}{M}}{}}{\begin{bmatrix}{1/2} & 0 & 0 \\0 & {1/2} & 0 \\0 & 0 & 0\end{bmatrix}}{\underset{\underset{P_{cav}}{}}{\begin{bmatrix}P_{{cav},x} \\P_{{cav},y} \\P_{{cav},z}\end{bmatrix}}.}}}} & (39)\end{matrix}$

Example D corresponds to FIG. 3D for a spherical cavity. This is similarto the previous example, but using spherical coordinates instead ofcylindrical. Now, we take θ as the polar angle and ϕ as the azimuthangle, then if E_(in)=E_(in){circumflex over (z)} we have the separationbetween charges of d₀=d₀ cos θ and the charges on the sphere q₀=q₀ cos θso that p₀=p₀ cos² {circumflex over (z)}. Therefore,

$\begin{matrix}{{\langle p\rangle} = {{\frac{1}{4\pi}{\int_{\varphi = 0}^{2\pi}{\int_{\theta = 0}^{\pi}{p_{0}\cos^{2}\varphi \sin \theta d\varphi d\theta}}}} = {\frac{( {p_{0}\hat{z}} )}{3}.}}} & (40)\end{matrix}$

The other symmetry axes have a similar calculation. Therefore,exploiting the symmetry of a sphere

$\begin{matrix}{E_{cav} = {E_{in} - {\frac{1}{\epsilon_{med}}\underset{\underset{\overset{=}{M}}{}}{\begin{bmatrix}{1/3} & 0 & 0 \\0 & {1/3} & 0 \\0 & 0 & {1/3}\end{bmatrix}}{\underset{\underset{P_{cav}}{}}{\begin{bmatrix}P_{{cav},x} \\P_{{cav},y} \\P_{{cav},z}\end{bmatrix}}.}}}} & (41)\end{matrix}$

Therefore, generalizing

$\begin{matrix}{E_{in} = {E_{cav} + \frac{\overset{=}{M} \cdot P_{cav}}{\epsilon_{med}}}} & (42)\end{matrix}$

so by Eq. 35 the second term should be E_(par), however in the particleP_(par)+P_(cav)=0

$\begin{matrix}{E_{in} = {E_{cav} - \frac{\overset{=}{M} \cdot P_{par}}{\epsilon_{med}}}} & (43)\end{matrix}$

therefore, in summary

$\begin{matrix}{E_{in} = {E_{par} + E_{cav}}} & (44) \\{E_{par} = {- \frac{\overset{=}{M} \cdot P_{par}}{\epsilon_{med}}}} & (45)\end{matrix}$

where M is the tensor depolarization factor in an arbitrary coordinatebasis. For the avoidance of doubt note that by dividing by ∈_(med) wehave backfilled the cavity with a medium that is not necessarily vacuumas was already discuss. Also, see Eqs. 147-150.

In the examples presented so far, the depolarization M was constructedin a coordinate basis providing a simple diagonal matrix for itsrepresentation. Going forward, more care is applied and the diagonalrepresentation will carry a single prime. So for example what has beencalled M above, i.e. a diagonal matrix basis, is more formallyrepresented as M in the eigen symmetry basis of the particle. Thisconvention is to be applied going forward in this document, unlessstated otherwise.

Additionally, in the observer's coordinates, we will write M, i.e.without a prime, as the representation of the depolarization tensor.This coordinate transformation transforms the particle into theappropriate direction in the observer's coordinate system and isdeveloped by considering the eigen-value problems Mw_(j)=M′_(jj) w_(j)for each coordinate j. We then find

[ Mw ₁ ′,Mw ₂ ′,Mw ₃′]=[M ₁₁ w ₁ ′,M ₂₂ w ₃ ′,M ₃₃ w ₁′]  (46)

and therefore

$\begin{matrix}{{\overset{=}{M}\lbrack {w_{1}^{\prime},w_{2}^{\prime},w_{3}^{\prime}} \rbrack} = {\lbrack {w_{1}^{\prime},w_{2}^{\prime},w_{3}^{\prime}} \rbrack \begin{bmatrix}M_{11}^{\prime} & 0 & 0 \\0 & M_{22}^{\prime} & 0 \\0 & 0 & M_{33}^{\prime}\end{bmatrix}}} & (47)\end{matrix}$

where the diagonal matrix is the depolarization tensor M′ in thesymmetry basis. Let the eigen-vector matrix, formed by (eigenvectorcolumn vectors) be called W=[w₁′, w₂′, w₃′] then M W=W M′ and theresulting similarity transformation converts depolarization in the eigensymmetry basis to the observer's basis by

M=W M′W ⁻¹.  (48)

Both active and passive transformations are sometimes needed in PSO. Forexample, an active rotation of a particle by an angle of: ψ about the z′axis as a twist, θ about the y′ axis as the spherical polar angle, and ϕabout the z′ axis again as the spherical azimuth angle, then

$\begin{matrix}{\overset{=}{W} = \lbrack \begin{matrix}{{\cos \; {\theta cos\psi cos\varphi}} - {\sin \; {\psi sin\varphi}}} & {{{- \cos}\; {\theta sin\psi cos\varphi}} - {\cos \; {\psi sin\varphi}}} & {\sin \; {\theta cos\varphi}} \\{{\cos \; {\theta cos\psi sin\varphi}} + {\sin \; {\psi cos\varphi}}} & {{\cos \; {\psi cos\varphi}} - {\cos \; {\theta sin\psi sin\varphi}}} & {\sin \; {\theta cos\varphi}} \\{{- \sin}\; {\theta cos\psi}} & {\sin \; {\theta sin\psi}} & {\cos \; \theta}\end{matrix} \rbrack} & (49)\end{matrix}$

and M is a unitary rotation matrix so that its inverse is also itstranspose W ⁻¹=W ^(T). Therefore, the induced field in the cavity isgiven by

$\begin{matrix}{E_{par} = \frac{\overset{=}{M} \cdot P_{par}}{\epsilon_{med}}} & (50) \\{\overset{=}{M} = {\overset{=}{W}{\overset{=}{M}}^{\prime}{{\overset{=}{W}}^{- 1}.}}} & (51)\end{matrix}$

where M is the tensor depolarization factor, in the observer'scoordinates, that accounts for the particle's shape. Additionally, thetrace of M is unity, is proved later (see Eq. 167) to be

M _(xx) +M _(yy) +M _(zz)=1.  (52)

Note that the trace of a matrix is the sum of its eigenvalues, andtherefore must be invariant with respect to a change of basis.

Some Important Definitions

The cavity field may be thought of as inducing the polarization of theparticle and vice versa, whereby

p _(par)=∈_(med) αE _(cav)  (53)

P _(par) =Np _(par)  (54)

P _(par)=∈_(med) χE _(in)  (55)

where a is the tensor polarizability that connects a particle's dipolemoment p_(par) to the cavity field ∈_(cav) and where X is the tensorelectric susceptibility that connects the polarization density P_(par)to the input field E_(in), ∈_(med) is the permittivity if the mediumsurrounding the particle i.e. vacuum, liquid, etc. Note that differentauthors may or may not include ∈_(med) in the definitions above so thereader must always check. The quantity N is discussed in the nextsub-section.

Particle Density Factors

The particle density N is the number of particles of a particular typein a given volume. If V_(par) is the volume of a single particle andV_(med)=V_(uc) is the volume of the surrounding medium contained in aeffective unit cell where only one particle exists on average then

$\begin{matrix}{N = {\frac{\# \mspace{14mu} {of}\mspace{14mu} {particles}}{{{Vol}.\mspace{14mu} {unit}}\mspace{14mu} {cell}} = \frac{1}{V}}} & (56)\end{matrix}$

where the total volume in the unit cell is V=V_(par)+V_(med). However,the volume fraction of the particle in the unit cell is

$\begin{matrix}{\nu_{par} = \frac{V_{par}}{V}} & (57)\end{matrix}$

therefore,

$\begin{matrix}{{N = \frac{\nu_{par}}{V_{par}}}.} & (58)\end{matrix}$

If we define the maximum particle density as

$\begin{matrix}{N_{\max} = \frac{1}{V_{par}}} & (59)\end{matrix}$

then we arrive at the relation

N=N _(max) v _(par)  (60)

where 0≤v_(par)≤1, where typically v_(par) is typically much less thanunity. For example, a random packing of spheres will generally have adensity of about 64% and regular face centered cubic packing of sphereswill have a maximum packing of about 74%. The maximum packing ofnon-spherical spheroids is still an open question in mathematics, butcurrently appears to be around 75%. In principle, it is anticipated that0≤v_(par)<2/3 for many NP geometries of interest and even less of arange in practice. One possible exception of interest is cubic orcubic-like NPs where v_(par)=1 is theoretically possible, but currentlymuch more difficult to manufacture than spheroidal NPs—e.g. spheres,prolate spheroids, and oblate spheroids. Finally, for historical reasonsin much of this document V_(par)=ν when discussing NPs subject to DEPelectrokinetics.

Complex Tensor Electric Susceptibility

It is critically important to find a relation between the dipole momentand the input electric field because induced forces, torques, andstresses on particles are directly related to the dipole moment of aparticle. By plugging Eq. 53 into Eq. 54 we find that

P _(par)=∈_(med) NαE _(cav)  (61)

which shows that to order that the polarization density is proportionalto the local field within the particle and that the direction of thepolarization density may not be in the direction of the local field.This relation is an extension of the case for scalar polarizability andhas proved quite accurate for most practical applications wherenon-linear effects are not observed. Next, observe that by using Eqs. 43we have that

$\begin{matrix}{{{E_{in} = {E_{cav} + E_{par}}}{= {E_{cav} - \frac{\overset{=}{M} \cdot P_{par}}{\epsilon_{med}}}}} = {{E_{cav} - \frac{\epsilon_{med}{\overset{=}{M} \cdot N}\; \overset{=}{\alpha}E_{cav}}{\epsilon_{med}}} = {\lbrack {\overset{=}{I} - {N\overset{=}{M}\; \overset{=}{\alpha}}} \rbrack E_{cav}}}} & (62)\end{matrix}$

Therefore,

E _(cav)=[ I−M (N α)]⁻¹ E _(in)  (63)

and from Eq. 53 we have

p _(par)=∈_(med) α[ I−M (N α)]⁻¹ E _(in)  (64)

However, for a single NP of volume ν we have that N=1/ν so that

p _(par)=∈_(med)ν(N α)[ I−M (N α)]⁻¹ E _(in).  (65)

and on dividing both sides by the particle's volume we get the inducedpolarization density of the single particle

$\begin{matrix}{P_{par} = {\epsilon_{med}\underset{\underset{{Electric}\mspace{14mu} {Susceptibility}\; \overset{=}{\chi}}{}}{{( {N\; \overset{=}{\alpha}} )\lbrack {\overset{=}{I} - {\overset{=}{M}\; ( {N\; \overset{=}{\alpha}} )}} \rbrack}^{- 1}}{E_{in}.}}} & (66)\end{matrix}$

This equation is really just Eq. 55. Equations 65-66 are very importantequations because they relate the input electric field to the induceddipole moment. Given its importance, let's write the unitlesscomplex-valued susceptibility tensor as

$\begin{matrix}{\overset{=}{\chi} = {{\overset{=}{\chi}}_{R} + {i{\overset{=}{\chi}}_{I}}}} & (67) \\{= {( {N\; \overset{=}{\alpha}} )\lbrack {\overset{=}{I} - {\overset{=}{M}\; ( {N\; \overset{=}{\alpha}} )}} \rbrack}^{- 1}} & (68)\end{matrix}$

where the real part is χ _(R) and the imaginary part is χ _(I). Theelectric susceptibility has real and imaginary parts because it isfrequency dependent due to the non-zero conductivity of the materials inthe NP and in the surrounding medium. Note that when there is a verysmall difference between the dielectric constant of the medium ∈_(med)and the material comprising the particle ∈ _(par) then the factor insquare brackets is very close to the identity matrix I and the CM factoris simply Nα. For example this is the case for blood cells in a liquidplasma. However, for many crystals, e.g. TiO₂, there is a largedielectric contrast compared to a surrounding liquid and the entireexpression as provided in Eq. 68 must be used. In summary

p _(par)=∈_(med) νχE _(in)  (69)

P _(par)=∈_(med) χE _(in)  (70)

χ=(N α)[ I−M (N α)]⁻¹.  (71)

Complex Tensor Polarizability

Next, equate the polarization density from Eq. 70 and 71 to thepolarization density from

D _(in)=∈_(med) E _(in) +P _(par)={dot over (∈)}_(par) E _(in)  (72)

whereby

P _(par)=∈_(med) χE _(in)=( E _(par) −I∈ _(med))E _(in)  (73)

so that

∈_(med)(Nα)[ I−M (N α)]⁻¹=∈ _(par) −I∈ _(med)  (74)

and therefore,

N α=[∈ _(par) M+∈ _(med)( I−M )]⁻¹(∈ _(par)− I ∈_(med))  (75)

This is an important relation that gives the polarizability α in termsof the depolarization factor A, the permittivity of the surroundingmedium ∈_(med), and the tensor permittivity of the medium comprising theparticle ∈ _(par). It should be pointed out that the built-in charges ofa crystal manifest themselves in ∈ _(par) and can be derivedindependently of the geometric considerations of the depolarizationfactor A.

Equation 75 will often be used in the particle's eigen symmetry basis.Thus, more specifically we can write it as

N α=[∈ _(par) ′M′+∈ _(med)( I−M ′)]⁻¹(∈ _(par)′− I ∈_(med))  (76)

Clausius-Mossotti & Lorentz-Lorenz Factors

Historically, the scalar electric susceptibility has been written with afactor that is without units and normalized to a range of +1 for thespherical particle. That factor is called the Clausius-Mossotti (CM)factor in the context of DEP and it called the Lorentz-Lorenz (LL)factor in the context of optics. The objective of this section is toextend the definition of the CM factor and LL factor for anisotropicparticle media.

Consider Eq. 71 and multiply through by M′, the result is called the CMtensor in the particle's eigen symmetry basis, and is given by

K′=M′χ′=M ′(N α′)[ I−M ′(N α′)]⁻¹,  (77)

and in the observer's basis

K=W K′W ⁻¹.  (78)

where each of the diagonal entries in M′ is a value between zero andone. For example for a sphere M′=Z/3.

When the elements of I−M′(Nα′)≈I then Eq. 77, e.g. a biological cell andthe surrounding liquid have on a small difference in permittivity andconductivity, then

K′=M′χ′≈M ′(N α′)  (79)

Ellipsoidal-Hyperbolic Coordinates

The spheroidal shape, an example of which is shown in FIG. 4C, isexceptional in its geometric ability to represent many NPs includingprolate spheroids, oblate spheroids, spheres, cylinders, and triaxialspheroids. Let's look at two dimensions before three dimensions for abetter understanding while using less math.

Therefore, consider FIG. 4A, which shows a generic ellipse. The ellipsehas major principle axis of length 2 a, a minor principle axis of length2 b, and distance between foci of length 2 c. The ellipse is defined bythe property that the sum of the distance from two foci to the curve isa constant κ₁. That is

${\overset{\_}{F_{1}P} + \overset{\_}{F_{2}P}} = {\kappa_{1}.}$

However, we can use the special case of point P coinciding with point V,so that

$\kappa_{1} = {{\overset{\_}{F_{1}P} + \overset{\_}{F_{1}P}} = {{( {a + c} ) + ( {a - c} )} = {2{a.}}}}$

Having found κ₁=2a we can write √{square root over((c+x)²+y²+(c−x)²+y²)}=2a so that after a bit of algebra

$\begin{matrix}{{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2} - c^{2}}} = {{1\mspace{14mu} {where}\mspace{14mu} a} > c}} & (80)\end{matrix}$

where we define b²=a²−c², which is consistent with the Pythagorustheorem. A similar situation exists for a hyperbolic curve in FIG. 4B,where

${\overset{\_}{F_{1}P} - \overset{\_}{F_{2}P}} = \kappa_{2}$

for point P=V we find that

$\kappa_{2} = {{\overset{\_}{F_{1}P} - \overset{\_}{F_{2}P}} = {{( {c + a} ) - ( {c - a} )} = {2{a.}}}}$

Therefore, √{square root over ((c+x)²+y²−(c−x)²+y²)}=2a and

$\begin{matrix}{{\frac{x^{2}}{a^{2}} - \frac{y^{2}}{c^{2} - a^{2}}} = {{1\mspace{14mu} {where}\mspace{14mu} a} < c}} & (81)\end{matrix}$

where we define b²=a²−c². A couple of important observations are inorder at this point. First, both equations can be condensed into oneequation system taking the form

$\begin{matrix}{{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}} = 1} & (82) \\{b^{2} = {a^{2} - c^{2}}} & (83)\end{matrix}$

where the sign of the second term in Eq. 82 determines if an ellipse ora hyperbola is represented by Eq 82. Second, for an ellipse andhyperbola we can write

$\begin{matrix}\begin{matrix}{c^{2} = {a^{2} - b^{2}}} \\{{= {\underset{A^{2}}{\underset{}{\lbrack {a^{2} + {f(\xi)}} \rbrack}} - \underset{B^{2}}{\underset{}{\lbrack {b^{2} + {f(\xi)}} \rbrack}}}},}\end{matrix} & (84)\end{matrix}$

where ƒ(ξ) is an arbitrary function of a new coordinate ξ. Thus we canfind another major axis A and another minor axis B such that C=c isconserved and the focal points are unchanged and confocal for both theellipse and hyperbola. All that is required is to add to the squaredlengths of the major and minor axes the same arbitrary function ƒ(ξ).The simplest example of of such a function is ƒ(ξ)=ξ. Therefore,following this line of reasoning a family of two-dimensionalelliptic-hyperbolic coordinates requires two equations parameterized bytwo new coordinates {ξ, η}.

If 0<b<a then the principle axis of the ellipse and hyperbola arex-directed, as shown in FIG. 4A-B and

$\begin{matrix}{{{x\text{-}{Ellipse}\text{:}\mspace{14mu} \frac{x^{2}}{a^{2} + \xi}} + \frac{y^{2}}{b^{2} + \xi}} = {{{{{1( {a^{2} + \xi} )} > 0}\&}( {b^{2} + \xi} )} > 0}} & (85) \\{{{x\text{-}{H{yperbola}}\text{:}\mspace{14mu} \frac{x^{2}}{a^{2} + \eta}} + \frac{y^{2}}{b^{2} + \eta}} = {{{{{1( {a^{2} + \eta} )} > 0}\&}( {b^{2} + \eta} )} < 0.}} & (86)\end{matrix}$

Alternately, if 0<a<b then the principle axis of the ellipse andhyperbola are y-directed and

$\begin{matrix}{{{x\text{-}{Ellipse}\text{:}\mspace{14mu} \frac{x^{2}}{a^{2} + \xi}} + \frac{y^{2}}{b^{2} + \xi}} = {{{{{1( {a^{2} + \xi} )} > 0}\&}( {b^{2} + \xi} )} > 0}} & (87) \\{{{x\text{-}{H{yperbola}}\text{:}\mspace{14mu} \frac{x^{2}}{a^{2} + \eta}} + \frac{y^{2}}{b^{2} + \eta}} = {{{{{1( {a^{2} + \eta} )} < 0}\&}( {b^{2} + \eta} )} > 0.}} & (88)\end{matrix}$

Furthermore, by solving for x and y we obtain

$\begin{matrix}{x = {\pm \sqrt{\frac{( {a^{2} + \xi} )( {\eta + a^{2}} )}{b^{2} - a^{2}}}}} & (89) \\{{y = {\pm \sqrt{\frac{( {b^{2} + \eta} )( {b^{2} + \xi} )}{b^{2} - a^{2}}}}},} & (90)\end{matrix}$

where only the positive (+) solutions are chosen in the first quadrantand the 4-fold symmetry of the coordinate system is exploited.

The above discussion is now extended to three dimensions as shown inFIG. 4C-F. In particular, consider three defining equations of theconfocal ellipsoidal coordinate system

$\begin{matrix}{\mspace{79mu} {{{Assuming}\text{:}\mspace{14mu} 0} < a < b < c < \infty}} & (91) \\{\mspace{79mu} {{{{E{llipsoide}}\text{:}\mspace{14mu} \frac{x^{2}}{a^{2} + \xi}} + \frac{y^{2}}{b^{2} + \xi} + \frac{z^{2}}{c^{2} + \xi}} = {1\mspace{31mu} \{ {\xi:{( {a^{2} + \xi} ) > 0}} \}}}} & (92) \\{{{{Hyperbola}\mspace{14mu} {of}\mspace{14mu} 1\text{-}{Sheet}\text{:}\mspace{14mu} \frac{x^{2}}{a^{2} + \eta}} + \frac{y^{2}}{b^{2} + \eta} + \frac{z^{2}}{c^{2} + \eta}} = {1\mspace{31mu} \{ {\eta:{( {b^{2} + \eta} ) < 0}} \}}} & (93) \\{{{{{Hyperbola}\mspace{14mu} {of}\mspace{14mu} 2\text{-}{{Sheet}s}\text{:}\mspace{14mu} \frac{x^{2}}{a^{2} + \zeta}} + \frac{y^{2}}{b^{2} + \zeta} + \frac{z^{2}}{c^{2} + \zeta}} = {1\mspace{31mu} \{ {\zeta:{( {c^{2} + \zeta} ) < 0}} \}}}\mspace{20mu}} & (94)\end{matrix}$

which is an asymmetric prolate spheroid with the longest axis along thez-axis, as shown in FIG. 4C, then

$\begin{matrix}{x = {\pm \sqrt{\frac{( {a^{2} + \xi} )( {a^{2} + \eta} )( {a^{2} + \zeta} )}{( {b^{2} - a^{2}} )( {c^{2} - a^{2}} )}}}} & (95) \\{y = {\pm \sqrt{\frac{( {b^{2} + \xi} )( {b^{2} + \eta} )( {b^{2} + \zeta} )}{( {a^{2} - b^{2}} )( {c^{2} - b^{2}} )}}}} & (96) \\{z = {\pm \sqrt{\frac{( {c^{2} + \xi} )( {c^{2} + \eta} )( {c^{2} + \zeta} )}{( {a^{2} - c^{2}} )( {b^{2} - c^{2}} )}}}} & (97)\end{matrix}$

where only the positive (+) solutions are chosen in the first octant andthe 8-fold symmetry of the coordinate system is exploited. The covariantbasis vectors are obtained from the position vector

r(ξ,η,ζ)=x(ξ,η,ξ){circumflex over (x)}+y(ξ,η,ζ)ŷ+z(ξ,η,ζ){circumflexover (z)}  (98)

so that the covariant basis vectors are

$\begin{matrix}{{e_{\xi} = \frac{\partial r}{\partial\xi}},\mspace{31mu} {e_{\eta} = \frac{\partial r}{\partial\eta}},\mspace{31mu} {e_{\zeta} = \frac{\partial r}{\partial\zeta}}} & (99)\end{matrix}$

and the metric tensor G=[e_(i)·e_(j)] shows that the basis vectors areorthogonal, but not normalized as seen here

$\begin{matrix}{{\mspace{855mu} {(100){\overset{=}{G} =}}\quad}{\quad{\lbrack \begin{matrix}\frac{( {\xi - \zeta} )( {\xi - \eta} )}{4( {a^{2} + \xi} )( {b^{2} + \xi} )( {c^{2} + \xi} )} & 0 & 0 \\0 & \frac{( {\eta - \zeta} )( {\eta - \xi} )}{4( {a^{2} + \eta} )( {b^{2} + \eta} )( {c^{2} + \eta} )} & 0 \\0 & 0 & \frac{( {\zeta - \eta} )( {\zeta - \xi} )}{4( {a^{2} + \zeta} )( {b^{2} + \zeta} )( {c^{2} + \xi} )}\end{matrix} \rbrack.}}} & \;\end{matrix}$

Next, define

R _(σ)=√{square root over ((a ²+σ)(b ²+σ)(c ²+σ))}  (101)

then the Lamé coefficients are defined as h_(k)=√{square root over(g_(kk))}

$\begin{matrix}{h_{\xi} = \frac{\sqrt{( {\xi - \zeta} )( {\xi - \eta} )}}{2R_{\xi}}} & (102) \\{h_{\eta} = \frac{\sqrt{( {\eta - \zeta} )( {\eta - \xi} )}}{2R_{\eta}}} & (103) \\{h_{\zeta} = \frac{\sqrt{( {\zeta - \eta} )( {\zeta - \xi} )}}{2R_{\zeta}}} & (104)\end{matrix}$

and the line element in terms of the Lamé coefficients is

(ds)² =h _(ξ)(dξ)² +h _(η)(dη)² +h _(ζ)(d _(ζ))².  (105)

Additionally, the gradient, divergence, and Laplacian can now be writtenin terms of the Lame coefficients as follows

$\begin{matrix}{\mspace{79mu} {{\nabla\Phi} = {{{\hat{e}}_{\xi}\frac{\partial\Phi}{h_{\xi}{\partial\xi}}} + {{\hat{e}}_{\eta}\frac{\partial\Phi}{h_{\eta}{\partial\eta}}} + {{\hat{e}}_{\zeta}\frac{\partial\Phi}{h_{\zeta}{\partial\zeta}}}}}} & (106) \\{\mspace{79mu} {{\nabla{\cdot A}} = {\frac{1}{h_{\xi}h_{\eta}h_{\zeta}}\lbrack {{\frac{\partial}{\partial\xi}( {h_{\eta}h_{\zeta}A_{\xi}} )} + {\frac{\partial}{\partial\eta}( {h_{\zeta}h_{\xi}A_{\eta}} )} + {\frac{\partial}{\partial\zeta}( {h_{\xi}h_{\eta}A_{\zeta}} )}} \rbrack}}} & (107) \\{{\nabla^{2}\Phi} = {\frac{1}{h_{\xi}h_{\eta}h_{\zeta}}\lbrack {{\frac{\partial}{\partial\xi}( {\frac{h_{\eta}h_{\zeta}}{h_{\xi}}\frac{\partial\Phi}{\partial\xi}} )} + {\frac{\partial}{\partial\eta}( {\frac{h_{\zeta}h_{\xi}}{h_{n}}\frac{\partial\Phi}{\partial\eta}} )} + {\frac{\partial}{\partial\zeta}( {\frac{h_{\xi}h_{\eta}}{h_{\zeta}}\frac{\partial\Phi}{\partial\zeta}} )}} \rbrack}} & (108)\end{matrix}$

where ê_(ξ)=e_(ξ)/|e_(ξ)|, ê_(η)=e_(η)/|e_(η)|, and ê_(ζ)=e_(ζ)/|e_(ζ)|.Finally, of particular interest is the resulting Laplacian in terms of{ξ, η, ζ} explicitly

$\begin{matrix}{{{\nabla^{2}\Phi} = {\frac{4}{( {\xi - \eta} )( {\xi - \zeta} )( {\eta - \zeta} )}\lbrack {{( {\eta - \zeta} )R_{\xi}\frac{\partial}{\partial\xi}( {R_{\xi}\frac{\partial\Phi}{\partial\xi}} )} + {( {\zeta - \xi} )R_{\eta}\frac{\partial}{\partial\eta}( {R_{\eta}\frac{\partial\Phi}{\partial\eta}} )} + {( {\xi - \eta} )R_{\zeta}\frac{\partial}{\partial\zeta}( {R_{\zeta}\frac{\partial\Phi}{\partial\zeta}} )}} \rbrack}},} & (109)\end{matrix}$

which is needed in determining the depolarization tensor for anellipsoidal NP.

The Spheroidal Conductor

In preparation for developing the depolarization tensor of a spheroidaldielectric it is instructive to first solve for the electric potentialin the space surrounding a conductive triaxial spheroid. In particular,assume that a NP is orientated and aligned in its primed symmetry basis.Then we can write the electric potential by leveraging Eqs. 95-97

$\begin{matrix}{\varphi_{{in},x^{\prime}} = {{{- E_{{in},x^{\prime}}}x^{\prime}} = {{- E_{{in},x^{\prime}}}\sqrt{\frac{( {a^{2} + \xi} )( {a^{2} + \eta} )( {a^{2} + \zeta} )}{( {b^{2} - a^{2}} )( {c^{2} - a^{2}} )}}}}} & (110) \\{\varphi_{{in},y^{\prime}} = {{{- E_{{in},y^{\prime}}}y^{\prime}} = {{- E_{{in},y^{\prime}}}\sqrt{\frac{( {b^{2} + \xi} )( {b^{2} + \eta} )( {b^{2} + \zeta} )}{( {a^{2} - b^{2}} )( {c^{2} - b^{2}} )}}}}} & (111) \\{\varphi_{{in},z^{\prime}} = {{{- E_{{in},z^{\prime}}}z^{\prime}} = {{- E_{{in},z^{\prime}}}{\sqrt{\frac{( {c^{2} + \xi} )( {c^{2} + \eta} )( {c^{2} + \zeta} )}{( {a^{2} - c^{2}} )( {b^{2} - c^{2}} )}}.}}}} & (112)\end{matrix}$

Clearly, in cartesian coordinates Laplace's equation ∇²Φ=0 is satisfiedseparately for each of the above equations. It is also satisfied in theellipsoidal-hyperbolic coordinates. Thus, it is easy to see that thesolutions are separable in {ξ, η, ζ}. For example, in the x′-direction

$\begin{matrix}{\varphi_{{in},x^{\prime}} = {\frac{- E_{in}}{\underset{\underset{C_{a\; 1}}{}}{\sqrt{( {b^{2} - a^{2}} )( {c^{2} - a^{2}} )}}}\underset{\underset{F_{a}{(\xi)}}{}}{\sqrt{a^{2} + \xi}}\underset{\underset{F_{a}{(\eta)}}{}}{\sqrt{a^{2} + \eta}}\underset{\underset{F_{a}{(\zeta)}}{}}{\sqrt{a^{2} + \zeta}}}} & (113)\end{matrix}$

and similar equations hold in the y′ and z′ directions. Therefore, ifF_(μ)(σ)=√{square root over (μ²+σ)} then

ϕ_(in,x′) =C _(a1) F _(a)(ξ)F _(a)(η)F _(a)(ζ)  (114)

ϕ_(in,y′) =C _(b1) F _(b)(ξ)F _(b)(η)F _(b)(ζ)  (115)

ϕ_(in,z′) =C _(c1) F _(c)(ξ)F _(c)(η)F _(c)(ζ)  (116)

However, while the potential on the surface of the conducting spheroidis zero, at infinity it is not zero according to Eqs 113-112. Thus,there must be another term to the solution that regularizes the solutionat infinity, i.e. when ι→∞, so that the solution ϕ→0 as ξ→∞. This isconsistent with Laplace's equation being a second order equation withtwo independent solutions. For example, in the x′-direction letF_(a)(ξ)+G_(a)(ξ) to regularize the electric potential at infinity, butleave the functionality otherwise unchanged. Then the particle'spotential in the medium surrounding the particle must take the form

ϕ_(med,z′) =C _(a2) G _(a)(ξ)F _(a)(η)F _(a)(ζ)  (117)

which in combination with Eq. 101 and Eq. 109 can be used to solveLaplace's equation ∇²ϕ_(med,x′)=0, whereby

$\begin{matrix}{{{R_{\xi}{\frac{d}{d\; \xi}\lbrack {R_{\xi}\frac{{dG}_{a}(\xi)}{d\; \xi}} \rbrack}} - {( {\frac{b^{2} + c^{2}}{4} - \frac{\xi}{2}} ){G_{a}(\xi)}}} = 0} & (118)\end{matrix}$

This is a 2^(nd) order ordinary differential equation that has one knownsolution G_(a1)=F_(a)(ξ) and one unknown solution G_(a2)(ξ), so that thecomplete solution is a superposition of G_(a)(ξ)=C_(a1) G_(a1)(ξ)+C_(a2)G_(a2)(ξ).

By way of review, the physicist and mathematician Jean le Rondd'Alembert (1717-1783) developed the method of reduction of order,whereby an equation having canonical form

$\begin{matrix}{{{\frac{d^{2}y}{{dx}^{2}} + {{p(x)}\frac{dy}{dx}} + {{q(x)}y}} = 0},} & (119)\end{matrix}$

having a solution y=y(x)=C₁ y₁(x)+C₂ y₂(x) can be solved if onlyy₁=y₁(x) is known. To find y₂=y₂(x) assume a product solution of theform y₂=v(x)y₁(x). Then on plugging this into Eq. 119 we find that

$\begin{matrix}{{\frac{d^{2}v}{{dx}^{2}} + {( {p - {\frac{2}{y_{1}}\frac{{dy}_{1}}{dx}}} )\frac{dv}{dx}}} = 0} & (120)\end{matrix}$

which does not have a term in v and is therefore trivial to solve byquadrature. To see this observe that if u=dv/dx the above equation isreduced in order and becomes

$\begin{matrix}{{\frac{du}{dx} + {( {p + {\frac{2}{y_{1}}\frac{{dy}_{1}}{dx}}} )u}} = 0.} & (121)\end{matrix}$

Thus, on solving for the second solution to the original problem we findthat

$\begin{matrix}{{y_{2}(x)} = {{y_{1}(x)}{\int{\frac{e^{- {\int{{p{(x)}}{dx}}}}}{y_{1}^{2}(x)}{{dx}.}}}}} & (122)\end{matrix}$

Therefore, for the current problem we have that

$\begin{matrix}{{{p(\xi)} = {{\frac{1}{R_{\xi}}\frac{{dR}_{\xi}}{d\; \xi}} = {\frac{d}{d\; \xi}\ln \mspace{14mu} R_{\xi}}}}{and}} & (123) \\{{G_{a\; 2}(\xi)} = {{{G_{a\; 1}(\xi)}{\int{\frac{e^{\int{{p{(\xi)}}d\; \xi}}}{G_{a\; 1}^{2}(\xi)}d\; \xi}}} = {{F_{a}(\xi)}{\int{\frac{d\; \xi}{{F_{a}^{2}(\xi)}R_{\xi}}.}}}}} & (124)\end{matrix}$

Thus, the two solutions to Eq. 118 are

$\begin{matrix}{{G_{a\; 1}(\xi)} = \sqrt{a^{2} + \xi}} & (125) \\{{G_{a\; 2}(\xi)} = {\sqrt{a^{2} + \xi}{\int_{\xi}^{\infty}\frac{d\; \sigma}{( {a^{2} + \sigma} )R_{\xi}}}}} & (126)\end{matrix}$

where R_(ξ) is given by Eq. 101. Similar equations for the otherprinciple directions are obtained by swapping a with b or c as needed.Also, σ is a dummy integration variable and the limits of integrationare chosen to regularize the solution at infinity. Now, taking theseresults and plugging back into Eq. 117 we find

$\begin{matrix}{\varphi_{{med},x^{\prime}} = {{C_{a\; 2}{F_{a}(\xi)}{F_{a}(\eta)}{F_{a}(\zeta)}{\int_{\xi}^{\infty}\frac{d\; \xi}{( {a^{2} + \sigma} )R_{\sigma}}}} = {\frac{\varphi_{{in},x^{\prime}}}{C_{a\; 1}}{\int_{\xi}^{\infty}\frac{d\; \xi}{( {a^{2} + \sigma} )R_{\sigma}}}}}} & (127)\end{matrix}$

The total electric potential is then

$\begin{matrix}{\varphi_{x^{\prime}} = {{\varphi_{{in},x^{\prime}} + \varphi_{{med},x^{\prime}}} = {{\varphi_{{in},x^{\prime}}\lbrack {1 + {\frac{C_{a\; 2}}{C_{a\; 1}}{\int_{\xi}^{\infty}\frac{d\; \sigma}{( {a^{2} + \sigma} )R_{\sigma}}}}} \rbrack}.}}} & (128)\end{matrix}$

To determine the ratio of constants note that the surface potentialϕ_(sur) must satisfy

$\begin{matrix}{{\varphi_{{sur},x^{\prime}} = {\varphi_{{in},x^{\prime}}\lbrack {1 + {\frac{C_{a\; 2}}{C_{a\; 1}}{\int_{0}^{\infty}\frac{d\; \sigma}{( {a^{2} + \sigma} )R_{\sigma}}}}} \rbrack}},} & (129)\end{matrix}$

so on solving for C_(a2)/C_(a1) and inserting into Eq. 128 we obtain

$\begin{matrix}{\varphi_{x^{\prime}} = {\varphi_{{in},x^{\prime}} + {( {\varphi_{{sur},x^{\prime}} - \varphi_{{in},x^{\prime}}} )\frac{\int_{\xi}^{\infty}\frac{d\; \sigma}{( {a^{2} + \sigma} )R_{\sigma}}}{\int_{0}^{\infty}\frac{d\; \sigma}{( {a^{2} + \sigma} )R_{\sigma}}}}}} & (130)\end{matrix}$

where the integrals in this equation are elliptic. The surface potentialϕ_(sur,x′) is obtained from the charge it holds and the capacitance ofthe spheroid, which is not calculated here. Analogous equations existfor the y′ and z′ directions by changing a² to b² and c² respectively aswell as the value of the input potential.

Spheroidal Depolarization of a Dielectric

Consider a dielectric NP that is isotropic in its material propertiesand formed as a prolate spheroid. Let us further assume that the NP onlysupports bound charges in the form of polarized dipole moments and hasno free charges. The NP is then subject to the continuity boundaryconditions: (1) the electric potential, and (2) the normal component ofthe displacement field in the absence of free charges.

To start let's assume that the input electric field is in the x′direction so the results of the prior section can be used. Additionally,let the electric potential of the medium outside the NP beϕ_(med)=ϕ_(med)(ξ, η, ζ) and the electric potential inside the particlebe ϕ_(par)=ϕ_(par)(ξ, η, ζ), then in terms of F, from the previoussection, we have for ξ≥0, i.e. at both the surface itself and outsidethe surface of the P that

$\begin{matrix}{{\varphi_{med}( {\xi,\eta,\zeta} )} = {{C_{1}{F_{a}(\xi)}{F_{b}(\eta)}{F_{c}(\zeta)}} + {C_{2}{F_{a}(\xi)}{F_{b}(\eta)}{F_{c}(\zeta)}{\int_{\xi}^{\infty}\frac{d\; \sigma}{( {a^{2} + \sigma} )R_{\sigma}}}}}} & (131)\end{matrix}$

where the first term tends to infinity at an infinite distance from theNP and zero at the NP origin; and the second term compensates by tendingto zero at infinity and a finite value at the NP surface. Also, insidethe NP (−a²<ξ≤0)

ϕ_(par)(ξ,η,ζ)=C ₃ F _(a)(ξ)F _(b)(η)F _(c)(ζ)  (132)

which only tends to zero at the origin of the NP. Therefore, at the NPboundary itself, i.e. when ξ=0, the continuity of the electric potentialand normal component of the displacement field requires

$\begin{matrix}{{\varphi_{par}( {0,\eta,\zeta} )} = {\varphi_{med}( {0,\eta,\zeta} )}} & (133) \\{{\frac{\epsilon_{par}}{h_{\xi}}\frac{\partial\varphi_{par}}{\partial\xi}( {0,\eta,\zeta} )} = {\frac{\epsilon_{med}}{h_{\xi}}\frac{\partial\varphi_{med}}{\partial\xi}( {0,\eta,\zeta} )}} & (134)\end{matrix}$

Reducing Eq. 133 we get

$\begin{matrix}{C_{3} = {C_{1} + {C_{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}}} & (135)\end{matrix}$

where R_(σ) is given by Eq. 101. Also, on reducing Eq. 134 we have that

$\begin{matrix}{{{\epsilon_{par}C_{3}\frac{{dF}_{a}(\xi)}{d\; \xi}} = {\epsilon_{med}{\frac{d}{d\; \xi}\lbrack {{C_{1}{F_{a}(\xi)}} + {C_{2}{F_{a}(\xi)}{\int_{\xi}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}} \rbrack}}}\ } & (136)\end{matrix}$

where F_(μ)(ν)=√{square root over (μ²+σ)} so that F(0)=a andF_(a)(0)=1/(2a). Additionally,

$\begin{matrix}{{{\frac{d}{d\; \xi}{\int_{\xi = 0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}} = \frac{- 1}{a^{3}{bc}}}\ } & (137)\end{matrix}$

then on using the chain rule on Eq. 136 and the facts just stated wearrive at

$\begin{matrix}{{C_{3}\epsilon_{par}} = {{\epsilon_{med}\lbrack {C_{1} - \frac{C_{2}}{abc} + {C_{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}} \rbrack}.}} & (138)\end{matrix}$

Finally, when we combine Eq. 135 with Eq. 138 we obtain

$\begin{matrix}{C_{2} = {{\frac{abc}{2}\lbrack \frac{\epsilon_{med} - \epsilon_{par}}{\epsilon_{med}} \rbrack}{C_{3}.}}} & (139)\end{matrix}$

Now to find the potential inside the NP take Eq. 135 and plug into C₃ inEq. 132. Then take C₂ in Eq. 139 and plug that into the resultingequation. Finally, take the definition of C₁ from Eq. 113 and use it toreduce the resulting equation. Then we find

$\begin{matrix}{C_{3} = \frac{C_{1}}{1 + {( \frac{\epsilon_{par} - \epsilon_{med}}{\epsilon_{med}} )\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}}} & (140)\end{matrix}$

Finally, take the definition of C₁ from Eq. 113 and exploit it to reducethe resulting equation even further, whereby

$\begin{matrix}{{\varphi_{{par},x^{\prime}}( {\xi,\eta,\zeta} )} = \frac{- {E_{{in},x^{\prime}}\lbrack \frac{{F_{a}(\xi)}{F_{b}(\eta)}{F_{c}(\zeta)}}{\sqrt{( {b^{2} - a^{2}} )( {c^{2} - a^{2}} )}} \rbrack}}{{1 + {( \frac{\epsilon_{par} - \epsilon_{med}}{\epsilon_{med}} )\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}}\ }} & (141)\end{matrix}$

where the quantity in brackets is identified from Eq. 113 as x′,therefore

$\begin{matrix}{{\varphi_{{par},x^{\prime}}( {x^{\prime},y^{\prime},z^{\prime}} )} = \frac{{- E_{{in},x^{\prime}}}x^{\prime}}{1 + {( \frac{\epsilon_{par} - \epsilon_{med}}{\epsilon_{med}} )\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}}} & (142)\end{matrix}$

and the magnitude of the electric field inside the & direction isdeduced from the negative gradient whereby the electric field in theparticle is

$\begin{matrix}{{E_{par}( {x^{\prime},y^{\prime},z^{\prime}} )} = {\frac{E_{{in},x^{\prime}}}{1 + {( \frac{\epsilon_{par} - \epsilon_{med}}{\epsilon_{med}} )\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}}.}} & (143)\end{matrix}$

Now rearranging this equation yields

$\begin{matrix}{E_{{in},x^{\prime}} = {E_{{par},x^{\prime}} + {{E_{{par},x^{\prime}}( \frac{\epsilon_{par} - \epsilon_{med}}{\epsilon_{med}} )}\frac{abc}{2}{\int_{0}^{\infty}{\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}.}}}}} & (144)\end{matrix}$

However, the displacement field isD_(par)=∈_(med)E_(par)+P_(par)=∈_(par)E_(par) so that in the x′direction

P _(par,x′)=(∈_(par)−∈_(med))∈_(par,x′)  (145)

and consequently

$\begin{matrix}{E_{{in},x^{\prime}} = {E_{{par},x^{\prime}} + {( \frac{P_{{par},x^{\prime}}}{\epsilon_{med}} )\frac{abc}{2}{\int_{0}^{\infty}{\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}.}}}}} & (146)\end{matrix}$

However, P_(par)+P_(cav)=0 and recalling Eq. 101 the above relation canbe written as

$\begin{matrix}{E_{{in},x^{\prime}} = {E_{{par},x^{\prime}} + E_{{cav},x^{\prime}}}} & (147) \\{E_{{cav},x^{\prime}} = {- \frac{M_{x^{\prime}x^{\prime}}P_{{cav},x^{\prime}}}{\epsilon_{med}}}} & (148) \\{M_{x^{\prime}x^{\prime}} = {\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )R_{\sigma}}}}} & (149) \\{{R_{\sigma} = \sqrt{( {a^{2} + \sigma} )( {b^{2} + \sigma} )( {c^{2} + \sigma} )}},} & (150)\end{matrix}$

which should be compared to Eqs. 44-45.

Moreover, by starting with Eq. 143 and multiplying both sides by(∈_(S)−∈_(med)) and then using Eq. 145 we obtain

$\begin{matrix}{P_{{par},x^{\prime}} = \frac{E_{{in},x^{\prime}}}{\frac{1}{\epsilon_{par} - \epsilon_{med}} + \frac{M_{x^{\prime}x^{\prime}}}{\epsilon_{med}}}} & (151)\end{matrix}$

Obviously, we have been only been considering the case of a NP in thesymmetry basis (eigen-basis) and at that only for an x′ directedelectric field excitation. By inspecting the equations for the x′direction we can generalize to the tensor form.

$\begin{matrix}{E_{cav} = {E_{in} + {\frac{1}{\epsilon_{med}}\underset{\underset{{\overset{\_}{\overset{\_}{M}}}^{\prime}}{}}{\begin{bmatrix}M_{x^{\prime}x^{\prime}} & 0 & 0 \\0 & M_{y^{\prime}y^{\prime}} & 0 \\0 & 0 & M_{z^{\prime}z^{\prime}}\end{bmatrix}}\underset{\underset{P_{par}}{}}{\begin{bmatrix}P_{{par},x} \\P_{{par},y} \\P_{{par},z}\end{bmatrix}}}}} & (152) \\{{M_{x^{\prime}x^{\prime}} = {\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + a^{2}} )\sqrt{( {\sigma + a^{2}} )( {\sigma + b^{2}} )( {\sigma + c^{2}} )}}}}}\ } & (153) \\{M_{y^{\prime}y^{\prime}} = {\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + b^{2}} )\sqrt{( {\sigma + a^{2}} )( {\sigma + b^{2}} )( {\sigma + c^{2}} )}}}}} & (154) \\{M_{z^{\prime}z^{\prime}} = {\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; \sigma}{( {\sigma + c^{2}} )\sqrt{( {\sigma + a^{2}} )( {\sigma + b^{2}} )( {\sigma + c^{2}} )}}}}} & (155) \\{P_{{par},x^{\prime}} = \frac{E_{{in},x^{\prime}}}{\frac{1}{\epsilon_{par} - \epsilon_{med}} + \frac{M_{x^{\prime}x^{\prime}}}{\epsilon_{med}}}} & (156) \\{P_{{par},y^{\prime}} = \frac{E_{{in},y^{\prime}}}{\frac{1}{\epsilon_{par} - \epsilon_{med}} + \frac{M_{y^{\prime}y^{\prime}}}{\epsilon_{med}}}} & (157) \\{P_{{par},z^{\prime}} = \frac{E_{{in},z^{\prime}}}{\frac{1}{\epsilon_{par} - \epsilon_{med}} + \frac{M_{z^{\prime}z^{\prime}}}{\epsilon_{med}}}} & (158)\end{matrix}$

or in more compact notation

$\begin{matrix}{{P_{par} = {{\epsilon_{m}\lbrack {{\epsilon_{par}{\overset{\_}{\overset{\_}{M}}}^{\prime}} + {\epsilon_{med}( {\overset{\_}{\overset{\_}{I}} - {\overset{\_}{\overset{\_}{M}}}^{\prime}} )}} \rbrack}^{- 1}( {\epsilon_{par} - \epsilon_{med}} )E_{in}}}{where}} & (159) \\{{\overset{\_}{\overset{\_}{M}}}^{\prime} = {\begin{bmatrix}M_{x^{\prime}x^{\prime}} & 0 & 0 \\0 & M_{y^{\prime}y^{\prime}} & 0 \\0 & 0 & M_{z^{\prime}z^{\prime}}\end{bmatrix}.}} & (160)\end{matrix}$

On comparison with Eq. 76 this can be written as

P _(par)×∈_(med)(N α′)E _(in)  (161)

where the particle itself comprises a material that is isotropic ∈_(par)=∈ _(par) I and similar expression for the surrounding medium.Thus, all the anisotropy is from the particle geometry. This particularderivation of the polarization density P_(par) from first principlesassumed that the NPs are far enough away form each other that the dipolefields from one NP do not appreciable interact with the dipole fields ofanother NP. However, if the NPs are close together then dipoleinteractions would occur and we would have an expression such as fromEq. 66, so that in the eigen symmetry basis

P _(par)=∈_(med) χ′E _(in),  (162)

where

{dot over (χ)}′=(N{dot over (α)}′)[İ−{dot over (M)}′(N α′)]⁻¹.  (163)

In summary,

$\begin{matrix}{P_{par} = \{ \begin{matrix}{{\epsilon_{med}( {N{\overset{\_}{\overset{\_}{\alpha}}}^{\prime}} )}E_{in}} & {{NPs}\mspace{14mu} {not}\mspace{14mu} {too}\mspace{14mu} {close}\mspace{14mu} {together}} \\{{{\epsilon_{med}( {N{\overset{\_}{\overset{\_}{\alpha}}}^{\prime}} )}\lbrack {\overset{\_}{\overset{\_}{I}} - {{\overset{\_}{\overset{\_}{M}}}^{\prime}( {N{\overset{\_}{\overset{\_}{\alpha}}}^{\prime}} )}} \rbrack}^{- 1}E_{in}} & {{NPs}\mspace{14mu} {very}\mspace{14mu} {close}\mspace{14mu} {together}}\end{matrix} } & (164)\end{matrix}$

It will be shown later that when the NPs are not too close together thenfrequency adjustable torques and frequency adjustable stability axisexist on colloid systems having ohmic losses to DEP fields. However,when NPs are close together then the torques go to zero, which is aphenomena that is called dipole shadowing. This must be accounted for inPSO designs by keeping the volume fraction of NP low enough to avoid thedipole shadowing yet high enough to control light efficiently. Thisdistinction in torque processes is developed directly from Eq. 164 indetail in subsequent sub-sections.

Additional mathematical properties of the depolarization factors in theeigen-basis can be ascertained as well. For example, the trace of M′ isthe same as the sum of the eigenvalues, which is a constant independentof the basis used. Therefore, we consider the sum

$\begin{matrix}{{M_{x^{\prime}x^{\prime}} + M_{y^{\prime}y^{\prime}} + M_{z^{\prime}z^{\prime}}} = {\frac{abc}{2}{\int_{0}^{\infty}{\lbrack {\frac{1}{\sigma + a^{2}} + \frac{1}{\sigma + b^{2}} + \frac{1}{\sigma + c^{2}}} \rbrack \frac{d\; \sigma}{\sqrt{( {\sigma + a^{2}} )( {\sigma + b^{2}} )( {\sigma + c^{2}} )}}}}}} & (165)\end{matrix}$

and take u=R_(σ) ²=(σ+a²)(σ+b²)(σ+c²). It is easy to see that

$\begin{matrix}{{\frac{1}{u}\frac{du}{d\sigma}} = {\frac{1}{\sigma + a^{2}} + \frac{1}{\sigma + b^{2}} + \frac{1}{\sigma + c^{2}}}} & (166)\end{matrix}$

so that

$\begin{matrix}{{M_{x^{\prime}x^{\prime}} + M_{y^{\prime}y^{\prime}} + M_{z^{\prime}z^{\prime}}} = {{\frac{abc}{2}{\int_{a^{2}b^{2}c^{2}}^{\infty}\frac{du}{u^{3/2}}}} = 1}} & (167)\end{matrix}$

Thus, as earlier examples suggested, in any cartesian basis we have thatthe sum of the eigen values of a depolarizations matrix is unity because

M _(x,x′) +M _(y,y′) +M _(z,z′)=1.  (168)

Next, consider the evaluation of the integrals in Eqs. 153-155.Evaluation of these depolarization-factor integrals are now provided forseveral important cases. The first case is for prolate triaxialspheroids where in general

$\begin{matrix}{M_{x^{\prime}x^{\prime}} = {( \frac{abc}{2} )\frac{\lbrack {\frac{{- 2}b}{a} + \frac{2c{E_{3}\lbrack {{\sin^{- 1}\lbrack \frac{\sqrt{c^{2} - a^{2}}}{c} \rbrack},\frac{c^{2} - b^{2}}{c^{2} - a^{2}}} \rbrack}}{\sqrt{c^{2} - a^{2}}}} \rbrack}{c( {a^{2} - b^{2}} )}}} & (169) \\{M_{y^{\prime}y^{\prime}} = {( \frac{abc}{2} )\frac{\lbrack {\frac{{- 2}a}{b} + \frac{2c{E_{3}\lbrack {{\sin^{- 1}\lbrack \frac{\sqrt{c^{2} - b^{2}}}{c} \rbrack},\frac{c^{2} - a^{2}}{c^{2} - b^{2}}} \rbrack}}{\sqrt{c^{2} - b^{2}}}} \rbrack}{c( {b^{2} - a^{2}} )}}} & (170) \\{M_{z^{\prime}z^{\prime}} = {( \frac{abc}{2} )\frac{\lbrack {\frac{{- 2}b}{c} + \frac{2a{E_{3}\lbrack {{\sin^{- 1}\lbrack \frac{\sqrt{a^{2} - c^{2}}}{a} \rbrack},\frac{a^{2} - b^{2}}{a^{2} - c^{2}}} \rbrack}}{\sqrt{a^{2} - c^{2}}}} \rbrack}{a( {c^{2} - b^{2}} )}}} & (171)\end{matrix}$

where ∈₃ is the elliptic function of the third kind, also called theelliptic-E function.

The second case for evaluating Eqs. 153-155 is for prolate and oblatespheroids defined by x′ and y′ axes having equal extent, i.e. a=b, wefind that M_(x,x′)=M_(y,y′) and if we define a new variable η=c/a thenthe depolarization factors are

$\begin{matrix}{M_{x^{\prime}x^{\prime}} = {\frac{1}{2} - \frac{1}{2( {1 - \eta^{2}} )} + \frac{\eta \cos^{- 1}\eta}{2( {1 - \eta^{2}} )^{3/2}}}} & (172) \\{M_{z^{\prime}z^{\prime}} = {\frac{1}{1 - \eta^{2}} - \frac{\eta \cos^{- 1}\eta}{( {1 - \eta^{2}} )^{3/2}}}} & (173) \\{{{2M_{x^{\prime}x^{\prime}}} + M_{z^{\prime}z^{\prime}}} = 1.} & (174)\end{matrix}$

Finally, as a check of principles consider the case of a sphere wherea=b=c then M_(x,x′)=M_(y,y′)=M_(z,z′) and we find that

$\begin{matrix}{M_{x^{\prime}x^{\prime}} = {{\frac{a^{3}}{2}{\int_{0}^{\infty}\frac{d\sigma}{( {\sigma + a^{2}} )\sqrt{( {\sigma + a^{2}} )( {\sigma + a^{2}} )( {\sigma + a^{2}} )}}}} = \frac{1}{3}}} & (175)\end{matrix}$

Which is the same value of M_(x′x′) as provided in the matrix of Eq. 41.The combined cases of the oblate spheroid, prolate spheroid, and sphereare provided in FIG. 5.

Therefore, if the input electric field is uniform and parallel in amedium E_(med) then the electric field within a particle (or cavity) isalso uniform, parallel, and independent of the orientation of the NP.Moreover E, is in general not parallel to E_(par) or E_(cav). Thiseffect is due entirely to the shape of the NP and its homogeneous,isotropic, and linear medium. Moreover, it should be noted that whilethe expressions for the depolarization factors cover many practicalsituations there are other examples, such as multi-layer particles thatare not covered in this solution. This in no way limits the general ideaand in fact it is possible to develop expressions or empirical studiesfor such cases as well.

Note that in the limit of substantially oblate, prolate, and sphericalspheroids (i.e. see FIG. 5) we have the approximations

$\begin{matrix}{{\overset{=}{M^{\prime}}({Oblate})} = \begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & 1\end{bmatrix}} & (176) \\{{\overset{=}{M^{\prime}}({Prolate})} = \begin{bmatrix}\frac{1}{2} & 0 & 0 \\0 & \frac{1}{2} & 0 \\0 & 0 & 0\end{bmatrix}} & (177) \\{{\overset{=}{M^{\prime}}({Sphere})} = \begin{bmatrix}\frac{1}{3} & 0 & 0 \\0 & \frac{1}{3} & 0 \\0 & 0 & \frac{1}{3}\end{bmatrix}} & (178)\end{matrix}$

and the resulting expressions that use the tensor depolarization factorsimplify substantially.

Connecting Depolarization to Particle and Cavity Fields

From Eqs. 42-45 and Eqs. 147-150 we have that

$\begin{matrix}{E_{in} = {E_{par} + E_{cav}}} & (179) \\{{P_{par} + P_{cav}} = 0} & (180) \\{E_{par} = {{- \frac{\overset{=}{M} \cdot P_{par}}{\epsilon_{m}}} = \frac{\overset{=}{M} \cdot P_{cav}}{\epsilon_{m}}}} & (181) \\{E_{cav} = {\frac{\overset{=}{M} \cdot P_{par}}{\epsilon_{m}} = {- \frac{\overset{=}{M} \cdot P_{cav}}{\epsilon_{m}}}}} & (182)\end{matrix}$

where it should be noted that a electrically neutral cavity and itscorresponding neutral particle have the same surface with oppositesigned charges. As the distance between a positive and negative chargein the particle is the same as that of the cavity then each elementaldipole moment of the cavity is of the same magnitude an opposite sign ofthe particle and Np_(cav)=−NP_(par) so that Eq. 180 results. Thus, thefour ways to connect E_(par) and E_(cav) to P_(par) and P_(cav) havebeen provided.

Polarizability and Susceptibility of Spheroids

The results of this sub-section are important because it connects thephysical parameters of the system to the polarizability andsusceptibility of spheroids, which are further connected to particleforces, torques, and stress. The relations developed below are used foractual calculations and to get a sense of how physical parameters impactthe performance of both particle electrokinetics and optics. Of greatimportance is the frequency response of colloidal (and other) mixtures.For example, in subsequent sections a frequency response, with ohmicloss, is shown to orient a particle in any desired direction by means ofharmonic electric field excitation, i.e. changing the frequency changesthe direction of the stable direction of particles. This is a keycapability for synthesizing optical systems. Note that the results ofthis sub-section assume that the constitutive parameters of permittivityand conductivity are a constant, i.e. not a function of frequency.Frequency is introduced directly via Maxwell's equations not a materialmodel such as the Lorentz and Drude models of a dielectric and metalrespectively.

The starting point is with the polarizability from Eq. 76

N α′=[∈ _(par) ′M′+∈ _(med)( I−M ′)]⁻¹(∈ _(par) ′−I∈ _(med)).  (183)

however, for an isotropic particle ∈ _(par)′=I∈_(par) we obtain,

N α′=[∈_(par) ′M′+∈ _(med)( I−M ′)]⁻¹(∈_(par)′−∈_(med)).  (184)

It is seen that the anisotropic nature of α′ is from the geometry of theparticle if the particle's material is isotropic; and any anisotropiccharacteristic is then given in the geometric depolarization factor M′.The components of the depolarization factor are given by Eqs. 172-173for the very practical symmetric spheroids, which are plotted in FIG. 5.Note that the restriction to isotropic materials is not necessary. If acrystal is used in the material making up particles then ∈ _(par)′ wouldnot in general be a diagonal matrix. It might of course still bepossible to diagonalize the resulting expression for the polarizabilityand susceptibility, however the eigen basis wold no longer have the samesimple interpretation that it does in the isotropic case.

The electric susceptibility in the particle's eigen symmetry basis isthen given by Eq. 71 as

χ′=(N α′)[ I−M ′(N α′)]⁻¹.  (185)

However, from Eq. 60 we know that N=N_(max)v_(par). Some discretion isrequired in using this expression for N in all parts of Eq. 185. Inparticular, the force or torque on a single particle will existindependent of the size of the medium surrounding a particle, butv_(par)→0 as the medium surrounding the single particle increases involume. As force and torque are proportional to χ′ the forces andtorques would be modeled as approaching zero when they are clearly notzero. To overcome this we make the following generalizing approximation

$\begin{matrix}{\overset{=}{\chi^{\prime}} \approx {{( {N\; \overset{=}{\alpha^{\prime}}} )\lbrack {\overset{=}{I} - {( \frac{\nu_{par}}{\nu_{\max}} ){\overset{=}{M^{\prime}}( {N\overset{=}{\alpha^{\prime}}} )}}} \rbrack}^{- 1}.}} & (186)\end{matrix}$

where as already discussed v_(max)≈2/3 is very roughly correct forrandom packed spheres and will be used as the baseline even for other NPshapes as well for a first order approximation.

Next observe that for the specific case of an isotropic particle mediumthe quantities Nα′, χ′, and K′ are all diagonal matrices. This allowsall the calculations can be simplified to scalar calculations. Startingwith each term of the diagonal in Eq. 184, which takes the form

$\begin{matrix}{{N\; \alpha^{\prime}} = \frac{\epsilon_{par} - \epsilon_{med}}{{\epsilon_{par}M^{\prime}} + {( {1 - M^{\prime}} )\epsilon_{med}}}} & (187)\end{matrix}$

were α′∈{α_(x′x′), α_(y′y′), α_(z′z′)} and M′∈{M_(x′x′), M_(y′y),M_(z′z′)}. Now in general

∈=∈_(R) −i∈ _(I)  (188)

σ=σ_(R) −iσ _(I)  (189)

however, the imaginary parts ∈_(I)≈0 and σ_(I)≈0 therefore, from Eq. 31and the assumption that the permittivity is far away from a materialresonance, as discussed in the Lorentz model of a dielectric, we havemedium and particle permittivities as

$\begin{matrix}{\epsilon_{med} = {\epsilon_{mR} - {i\frac{\sigma_{mR}}{\omega}}}} & (190) \\{\epsilon_{par} = {\epsilon_{pR} - {i{\frac{\sigma_{pR}}{\omega}.}}}} & (191)\end{matrix}$

Using Eq. 186 and plugging in Eqs. 190-191 we obtain

$\begin{matrix}{\chi^{\prime} \approx {\frac{K_{\infty}}{1 - {\beta \; M^{\prime}K_{\infty}}} + \frac{\frac{K_{0}}{1 - {\beta \; M^{\prime}K_{\infty}}} - \frac{K_{\infty}}{1 - {\beta \; M^{\prime}K_{\infty}}}}{1 + {\xi^{2}\tau_{MW}^{2}\omega^{2}}} + {i\frac{\lbrack \frac{K_{\infty} - K_{0}}{( {1 - {\beta \; M^{\prime}K_{0}}} )^{2}} \rbrack \tau_{MW}\omega}{1 + {\xi^{2}\tau_{MW}^{2}\omega^{2}}}}}} & (192)\end{matrix}$

where a new parameter β called the compactness is introduced. Inprinciple the compactness is a tensor and has components {dot over(β)}={β_(x′x′),β_(y′y′), β_(z′z′)}, however to keep the modeling simplefor now it is kept as a scalar consistent with a sphere. The resultingparameters of the electric susceptibility are then

$\begin{matrix}{\beta = {\frac{\nu_{par}}{\nu_{\max}} \approx {( \frac{3}{2} )\nu_{par}}}} & (193) \\{K_{0} = \frac{\sigma_{pR} - \sigma_{mR}}{{M^{\prime}\sigma_{pR}} + {( {1 - M^{\prime}} )\sigma_{mR}}}} & (194) \\{K_{\infty} = \frac{\epsilon_{pR} - \epsilon_{mR}}{{M^{\prime}\epsilon_{pR}} + {( {1 - M^{\prime}} )\epsilon_{mR}}}} & (195) \\{\tau_{MW} = \frac{{M^{\prime}\epsilon_{pR}} + {( {1 - M^{\prime}} )\epsilon_{mR}}}{{M^{\prime}\sigma_{pR}} + {( {1 - M^{\prime}} )\sigma_{mR}}}} & (196) \\{\xi = \frac{1 - {K_{\infty}\beta \; M^{\prime}}}{1 - {K_{0}\beta \; M^{\prime}}}} & (197) \\{{\xi \tau_{MW}} = \frac{{\epsilon_{mR}( {1 - {M^{\prime}( {1 - \beta} )}} )} + {{M^{\prime}( {1 - \beta} )}\epsilon_{pR}}}{{\sigma_{mR}( {1 - {M^{\prime}( {1 - \beta} )}} )} + {{M^{\prime}( {1 - \beta} )}\sigma_{pR}}}} & (198)\end{matrix}$

The case when β=0, i.e. when neighboring particles are far away provides

$\begin{matrix}{{\chi^{\prime} \approx {N\; \alpha^{\prime}}} = {\underset{\underset{\chi_{R}^{\prime}}{}}{K_{\infty} + \frac{K_{0} - K_{\infty}}{1 + {\tau_{MW}^{2}\omega^{2}}}} + {i{\underset{\underset{\chi_{I}^{\prime}}{}}{\frac{\lbrack {K_{\infty} - K_{0}} \rbrack \tau_{MW}\omega}{1 + {\tau_{MW}^{2}\omega^{2}}}}.}}}} & (199)\end{matrix}$

which is a function of M′ through K₀, K_(∞), and ξ_(MW). Next, the casewhen β=1 with neighboring particles close together provides

$\begin{matrix}{{\chi^{\prime} \approx {\underset{\underset{\chi_{R}^{\prime}}{}}{\frac{\epsilon_{pR} - \epsilon_{mR}}{\epsilon_{mR}} + \frac{\frac{\sigma_{pR}}{\sigma_{mR}} - \frac{\epsilon_{pR}}{\epsilon_{mR}}}{1 + {( \frac{\epsilon_{mR}}{\sigma_{mR}} )^{2}\omega^{2}}}} + {i\underset{\underset{\chi_{I}^{\prime}}{}}{\frac{( \frac{{\epsilon_{pR}\sigma_{mR}} - {\epsilon_{mR}\sigma_{pR}}}{\sigma_{mR}^{2}} )\omega}{1 + {( \frac{\epsilon_{mR}}{\sigma_{mR}} )^{2}\omega^{2}}}}}}},} & (200)\end{matrix}$

which is completely independent of the depolarization factor M′.Intermediate values of 0 produce intermediate results and the concept of“close” also depends on the shape of the NPs so that that above is onlyan approximation. Thus, all the factors of M′ cancel out when particlesare close to neighbors, but not when they are far away. This is acritical insight into why particles that are close together experiencepolarization shadowing, which reduces and ultimately eliminates torquesfor very closely packed particles energized with electric fields ofspecific phase in each principle direction of a NP—described in asubsequent section. This sets up an optimal range for v_(par) formaximum performance in controlling light scattering and DEP. This isdiscussed in greater detail in a subsequent sub-section.

Hybrid States of Matter

Thus, from the previous sub-section, colloids and suspensions ofspheroidal (ellipsoidal) particles can form a hybrid state of matter. Bychanging the distance between particles by DEP the volume fractionchanges the compactness parameter β from zero when particles are faraway from each other to unity when they are very close (touching). Atβ=1 the shape of the molecules becomes irrelevant as all thedepolarization factors M′ cancel out by dipole shadowing and particlerotation from DEP induced torques becomes impossible for electric fieldshaving a zero phase difference between each of the principle directions.This results in a “liquid glass state” where particles cluster to formregions having similar orientation. Therefore, by electronicallycontrollable DEP the state of matter that is somewhere between liquidand solid is controlled. The dipole shadowing effect can be used tolock-in or park a particular configuration of NP orientations so thatangular drift is minimized or eliminated.

Combined Lorentz & Drude Model of a Material

In the previous section it was assumed that parameters like conductivityand permittivity are constants. This is not always the case nearmaterial resonances and the expressions for polarizability andsusceptibility can become much more complex and offer an ever greaterrichness of phenomena. It is with this in mind that this section, whichincludes material that is typically considered well-known, is providedas a substantive reminder of the impact of material resonances.

The techniques of this section are applicable across the entirefrequency range, i.e. from DEP frequencies to optical frequencybandwidths. That said, the material response at DEP frequencies alsodepends on the material response at optical frequencies. Therefore,understanding and calculating the permittivity in Eq. 75 for a DEPprocess requires an encompassing theory across all frequencies.

In this section a simple one-dimensional model is provided thatcorresponds to one of the eigen symmetry basis directions in aparticle's tensor permittivity ∈. There are several potential dipolesources. First, it has an electronic component that is due to anelectron cloud being distorted around atoms that are under the influenceof an external electric field. Second, molecules and charges therein canalso distort chemical bonds when under the influence of an externalelectric field. Third, the distortions having both an external force anda restoring force can have multiple vibration modes, each of whichcontributes to the permittivity. Fourth, there are also thermal forcesthat give rise to a an average permittivity in a medium filled withpolar molecules. Moreover, the crystal geometry comprising a Bravaislattice and a basis set of atoms impacts the character of theatom-to-atom and molecule to molecule interactions.

It is well known that one can model an oscillating charge using a simpleNewtonian approach. The results are almost exactly the same result asthat from quantum electrodynamics. Therefore, by way of review we canmodel an atom with a charge cloud by a spring model that includes aforcing function from a driving electric field, a restoring forcemodeled as a spring with spring constant, and a loss mechanism that isassociated with interactions with neighboring atoms that can take awayenergy. In particular, if the displacement of an electron is x=x(t) then

$\begin{matrix}{{{m_{e}\frac{d^{2}{x(t)}}{dt^{2}}} + {m_{e}\gamma \frac{d{x(t)}}{dt}} + {m_{e}\omega_{0}^{2}{x(t)}}} = {qE_{in}e^{i\omega t}}} & (201)\end{matrix}$

where the restoring forces has a “spring constant” of m_(e)ω₀ ².Therefore, assuming that x(t)=x₀e^(iωt) we have

$\begin{matrix}{{x_{0} = \frac{q{E_{in}/m_{e}}}{{- \omega^{2}} + {i\gamma \omega} + \omega_{0}^{2}}}.} & (202)\end{matrix}$

and the polarization density is P=Np=Nqx₀, so that

$\begin{matrix}{{P = \frac{Nq^{2}{E_{in}/m_{e}}}{\omega_{0}^{2} - \omega^{2} + {i\gamma \omega}}},} & (203)\end{matrix}$

but P=∈₀χE_(in) therefore

$\begin{matrix}{\chi = \frac{N{q^{2}/( {m_{e}\epsilon_{0}} )}}{\omega_{0}^{2} - \omega^{2} + {i\gamma \omega}}} & (204)\end{matrix}$

However, first P=(∈−∈₀)∈_(in)=∈₀χE_(in) so the relative permittivity is∈_(r)=1+χ; and second there are a multitude of resonances as alreadydiscussed. Finally, identifying the plasma frequency asω_(pls)=Nq²/(m_(∈)∈₀) then

$\begin{matrix}{\epsilon_{r} = {1 + {\sum\limits_{j}{\frac{\omega_{pls}^{2}}{\omega_{0,j}^{2} - \omega^{2} + {i\gamma_{j}\omega}}.}}}} & (205)\end{matrix}$

This equation can be further broken down into a real and imaginary part∈_(r)=∈_(R)−i∈_(I) where

$\begin{matrix}{\epsilon_{R} = {1 + {\sum\limits_{j}\frac{\omega_{pls}^{2}( {\omega_{0_{j}}^{2} - \omega^{2}} )}{( {\omega_{0,j}^{2} - \omega^{2}} )^{2} + {\gamma_{j}^{2}\omega^{2}}}}}} & (206) \\{\epsilon_{I} = {\sum\limits_{j}{\frac{\omega_{pls}^{2}( {\gamma_{j}\omega} )}{( {\omega_{0,j}^{2} - \omega^{2}} )^{2} + {\gamma_{j}^{2}\omega^{2}}}.}}} & (207)\end{matrix}$

Theses equations are valid for both dielectrics and metals. When amaterial is a metal simply set ω=0. For example, setting ω₀=0 Eq. 205becomes

$\begin{matrix}{\epsilon_{r} = {1 - \frac{\omega_{pls}^{2}}{\omega^{2} - {i\gamma \omega}}}} & (208) \\{\mspace{20mu} {= {1 - \frac{\omega_{pls}^{2}/\omega}{\omega - {i\gamma}}}}} & (209) \\{\mspace{20mu} {= {1 - {\omega_{pls}^{2}{\tau^{2}( \frac{1 + {i/( {\omega \tau} )}}{1 + {\omega^{2}\tau^{2}}} )}}}}} & (210)\end{matrix}$

where τ=1/γ is the mean scattering time of the electrons. One importantpoint being that through the permittivity of dielectrics and metals onecan find the polarizability {dot over (α)} and susceptibility χ so thata non-zero induced dipole moment p_(ind) exists. Thus, via Eqs. 33-34both DEP forces and torques exist for both dielectrics as well asmetals. Moreover, both dielectrics and metals can scatter light andprovide a means for controlling light in PSO.

Another important point being that the conductivity is also complex.Recall that P=(∈_(r)−1)∈₀E=Np. Then in the frequency domain we can takethe derivative by multiplying by iω therefore the current is given by

$\begin{matrix}{j = {{( {i\omega} )P}\mspace{14mu} = {{i{\omega ( \frac{{- \omega_{pls}^{2}}/\omega}{\omega - {i\gamma}} )}E_{in}}\mspace{14mu} = {\frac{\omega_{pls}^{2}\tau E_{in}}{1 - {i\omega \tau}}\mspace{20mu} = {\sigma \; E}}}}} & (211)\end{matrix}$

where τ=1/γ. Therefore, the conductivity takes the form

$\begin{matrix}{{\sigma = {\sigma_{R} - {i\sigma_{I}}}}{where}} & (212) \\{\sigma_{R}\  = \ \frac{\omega_{pls}^{2}\tau}{1 + ( {\omega \tau} )^{2}}} & (213) \\{\sigma_{I}\  = \ \frac{\omega_{pls}^{2}{\tau ( {\omega \tau} )}}{1 + ( {\omega \tau} )^{2}}} & (214)\end{matrix}$

The real part of the conductivity corresponds to a conductive currentand the imaginary part corresponds to a displacement current.

Tensor Permittivity of a Particle

If we consider Eq. 205 as the general one-dimensional express for therelative permittivity in one of the eigen symmetry basis directions thenwe can extend the analysis to anisotropic media as follows. In the eigensymmetry basis (represented by double primes) of an underlying crystalforming a NP we can use equations like Eq. 205 along the diagonal of apermittivity tensor, whereby

$\begin{matrix}{{\overset{¯}{\overset{¯}{\epsilon}}}_{par}^{''} = {\begin{bmatrix}\epsilon_{11}^{''} & 0 & 0 \\0 & \epsilon_{22}^{''} & 0 \\0 & 0 & \epsilon_{33}^{''}\end{bmatrix}.}} & (215)\end{matrix}$

Physically this says that along the v₁ direction the relativepermittivity is ∈₁₁″. Along the v₂′ direction the relative permittivityis ∈₂₂″. Along the v₃′ direction the relative permittivity is ∈₃₃″.Notice that the directions have only a single prime on them because theyare measured with respect to the primed system, that is the principleaxis of the particle's shape. Said another way, the directions ofcrystal symmetry are measured relative to the coordinate used to specifythe geometric principle axis of the particle, which itself comprises acrystal material with a different set of symmetry directions due to thecrystallographic geometry.

Equation 215 can be transformed from the eigen symmetry basis of thecrystal into the eigen symmetry basis of the particle by atransformation. We do this by considering the eigenvalue problems

∈ _(par) ′v ₁′=∈₁₁ ″v ₁′  (216)

∈ _(par) ′v ₂′=∈₂₂ ″v ₂′  (217)

∈ _(par) ′v ₃′=∈₃₃ ″v ₃′  (218)

These three eigenvalue problems can be rearranged into a matrix format

∈ _(par)′[v ₁ ′,v ₂ ′,v ₃′]=[∈₁₁ ″v ₁′,∈₂₂ ″v ₂′,∈₃₃ ″v ₃′]  (219)

so that

∈_(par) ′V′=V′∈ _(par)″  (220)

and finally

∈_(par) ′=V′∈ _(par)″( V ′)⁻¹.  (221)

This equation transfers the relative permittivity into the coordinatesystem of the symmetry basis of a particle. The matrix ∈ _(par)″ isdiagonal and therefore symmetric even when the individual entries alongthe diagonal are complex numbers. The similarity transformation willtherefore result in a symmetric matrix ∈_(par)′. This is critical as itis consistent with electromagnetic reciprocity, which is a built-infeature of Maxwell's equations for reciprocal media. Additionally, thetransformation matrix V′ corresponds to a passive transformation whereinthe coordinates are changed, but the crystallographic symmetrydirections {v₁′, v₂′, v₃′} remain fixed in space. In this way, settingthe vectors {v₁′, v₂′, v₃′} fixes the crystallographic symmetrydirections relative the particle in the primed system.

Tensor Permittivity as a State Function

The permittivity of a particle is a thermodynamic state function withindependent variables of temperature, pressure, electric field, magneticfield, stress, strain, frequency and other parameters. Therefore, lightscattering is affected by these thermodynamic signals and suchthermodynamic processes are included herein without constraint. Thereare applications that can exploit these state variables.

The Average Polarizability

If there are several types of particles in the system then an averagevalue of the polarizability can be found. For example, imagine thatthere are w₁ NPs of one kind, w₂ NPs of a second kind, and w₃ moleculesof a liquid forming a colloid. Then the average polarizability iscalculated as a weighted sum

$\begin{matrix}{{\langle\overset{\_}{\overset{\_}{\alpha}}\rangle} = {\frac{{w_{1}{\overset{\_}{\overset{\_}{\alpha}}}_{1}} + {w_{2}{\overset{\_}{\overset{\_}{\alpha}}}_{2}} + {W_{3}{\overset{\_}{\overset{\_}{\alpha}}}_{3}}}{w_{1} + w_{2} + w_{3}}.}} & (222)\end{matrix}$

However, the ratio w₁/(w₁+w₂+₃) is simply the average number ofparticles, which can be given in terms of Eq. 60, which provides thatN=N_(max) v_(par)=N₁ v₁ and similar for the other components of the sum.Therefore,

N α

=v₁(N ₁ α ₁)+v ₂(N ₂ α ₂)+v ₃(N ₃ α ₃).  (223)

Equation 223 is a tensor form of the Lorentz-Lorenz mixing equation.Generalizing to different colloid properties X_(j) with values ξ_(j) isdone by introducing a continuous joint probability distribution so that

N α

=∫∫ . . . ∫ƒ_(X) ₁ _(,X) ₂ _(, . . . ,X) _(σ) (ξ₁,ξ₂, . . . ,ξ_(σ))(Nα)dξ ₁ dξ ₂ . . . dξ _(σ)  (224)

where ƒ_(X) ₁ _(,x) ₂ _(, . . .) (ξ₁, ξ₂, . . . , ξ_(σ)) is the jointprobability distribution function for properties {X₁, X₂, . . . ,X_(σ)}. This equation is very useful for ascertaining the effectivepolarizability when a complex material has probability distributions ofparticle volume fractions, particle shapes, sizes, particleorientations, Bravais lattice orientation, and crystalographic basis.This mixing of particles in a medium provides a way to develop usefulnew colloid properties.

The Scalar Lorentz-Lorenz Equation

When many atoms are packed close together, as is the case in a glass orsimilar optical material, then the medium external to the atoms isvacuum and the background medium is vacuum so that ∈=∈_(r)∈₀=n²∈₀, wheren is the RI of the optical material. If the atom is modeled as a spherethen we get the Lorentz-Lorenz expression from Eq. 175 and Eq. 184

$\begin{matrix}{{\frac{N\alpha}{3} = \lbrack \frac{n^{2} - 1}{n^{2} + 2} \rbrack}.} & (225)\end{matrix}$

The Scalar Lorentz-Lorenz Mixing Equation

To find what happens to the RI when two types of materials are mixedtogether in such a way as to assure that there is intimate contact withthe atomic scale materials we use a two component form of Eq. 223 aswell as Eq. 225, whereby

$\begin{matrix}{\frac{n^{2} - 1}{n^{2} + 2} = {{v_{1}( \frac{n_{1}^{2} - 1}{n_{1}^{2} + 2} )} + {v_{2}( \frac{n_{2}^{2} - 1}{n_{2}^{2} + 2} )}}} & (226)\end{matrix}$

which is the classical Lorentz-Lorenz mixing equation. This equation isvalid for PSO because the liquid and the larger particles are inintimate contact at the atomic scale.

The Scalar Linear Mixing Equation

By expanding Eq. 226 in a Taylor expansion about n=1 we find that

$\begin{matrix}{{\frac{n^{2} - 1}{n^{2} + 2} = {\frac{2( {n - 1} )}{3} - {\frac{1}{9}( {n - 1} )^{2}} - {\frac{4}{27}( {n - 1} )^{3}} + {\frac{11}{81}( {n - 1} )^{4}} -}}\mspace{11mu} \ldots} & (227)\end{matrix}$

and similar equations for n₁ and n₂ so that on retaining only the firstorder terms and plugging into Eq. 226 we find

n+v ₁ +v ₂=1+v ₁ n ₁ +v ₂ n ₂,  (228)

but the volume fractions add to unity v₁+v₂=1. Therefore

n=v ₁ n ₁ +v ₂ n ₂,  (229)

which is the linear mixing equation for a two-component optical mixturesuch as a simple colloid. It is valid when the constituent elements areatoms, molecules, and NPs that are much smaller than the wavelength oflight. Additionally, because the volume fractions add to unity v₁+v₂=1it is also true that for a colloid with n₁=n_(L) as the liquid RI, andn₂=n_(P) as the particle RI and similarly for the volume fractions then

n=n _(L)+(n _(P) −n _(L))v _(P)  (230)

so that the average RI of spherical particles is given in terms of theparticle volume fraction as indicated.

The Tensor Linear Mixing Equations

Equation 230 only provides the RI in the case of spherical particlesthat are not made of an anisotropic material or shape. To includenon-spherical particles and isotropic material we can utilize Eq. 76 fora simple colloid whereby

N α

=v_(L)(N _(L) a _(L))+v _(P)(N _(P) α _(P))  (231)

where the volume fractions of the liquid v_(L) and the particles v_(P)satisfy v_(L)+v_(P)=1. However, for optics Eq. 231 must incorporate thatlight travels in a vacuum between the atomic and molecular particles ofa liquid and solid. Therefore, at this spatial scale ∈_(med)=∈₀. If wefurther assume that the material comprising a NP is made from anisotropic material, without an anistropic crystal structure, then∈_(par)=∈₀∈_(P) I so that in a NP's eigen symmetry basis we have that

$\begin{matrix}\begin{matrix}{{N\; {\overset{\_}{\overset{\_}{\alpha}}}^{\prime}} = {\lbrack {{{\overset{\_}{\overset{\_}{\epsilon}}}_{par}^{\prime}{\overset{\_}{\overset{\_}{M}}}^{\prime}} + {\epsilon_{med}( {\overset{\_}{\overset{\_}{I}} - {\overset{\_}{\overset{\_}{M}}}^{\prime}} )}} \rbrack^{- 1}( {{\overset{\_}{\overset{\_}{\epsilon}}}_{par}^{\prime} - {\overset{\_}{\overset{\_}{I}}\epsilon_{med}}} )}} \\{= \begin{bmatrix}\frac{\epsilon_{P} - 1}{{\epsilon_{P}M_{x^{\prime}x^{\prime}}} + ( {1 - M_{x^{\prime}x^{\prime}}} )} & 0 & 0 \\0 & \frac{\epsilon_{P} - 1}{{\epsilon_{P}M_{y^{\prime}y^{\prime}}} + ( {1 - M_{y^{\prime}y^{\prime}}} )} & 0 \\0 & 0 & \frac{\epsilon_{P} - 1}{{\epsilon_{P}M_{z^{\prime}z^{\prime}}} + ( {1 - M_{z^{\prime}z^{\prime}}} )}\end{bmatrix}}\end{matrix} & (232)\end{matrix}$

and only particle shape influences light scattering. Thus, each diagonalterm takes the form

$\begin{matrix}{{N\; \alpha_{mm}^{\prime}} = {\frac{\epsilon_{P} - 1}{\epsilon_{P}{M_{m^{\prime}m^{\prime}}/{+ ( {1 - M_{m^{\prime}m^{\prime}}} )}}}.}} & (233)\end{matrix}$

For the sake of simplifying notation, let's temporarily set M_(m′m′)→Mthen each diagonal element of the eigen symmetry basis of Eq. 231 mustsatisfy

$\begin{matrix}{\frac{n^{2} - 1}{{n^{2}M} + 1 - M} = {{( \frac{n_{L}^{2} - 1}{{n_{L}^{2}M_{L}} + 1 - M_{L}} )v_{L}} + {( \frac{n_{p}^{2} - 1}{{n_{p}^{2}M_{p}} + 1 - M_{p}} )v_{p}}}} & (234)\end{matrix}$

where n²=∈, n_(L) ²=∈_(P), and n²=∈_(P). However, on the left side ofthe equation both ∈_(P) and M are unknown. Now observe to first orderthe following quantities

$\begin{matrix}{\frac{n^{2} - 1}{{n^{2}M} + 1 - M} = {{2( {N - 1} )} + \cdots}} & (235) \\{\frac{n^{2}}{{n^{2}M} + 1 - M} = {1 + {2( {1 - M} )( {n - 1} )} + \cdots}} & (236) \\{\frac{1}{{n^{2}M} + 1 - M} = {1 - {2{M( {n - 1} )}} + \cdots}} & (237)\end{matrix}$

and similar expressions for the ML and MP terms of Eq. 234. Note that tofirst order Eq. 235 has no terms that include M. This makes orientationanalysis extremely difficult. To make a proper first order theory wemust have additional insight on where the factors of M are in the lineartheory. One mathematical insight is that Eq. 234 is actually twoseparate equations that have been added together and the first orderfactors of M cancel out. As Eq. 234 needs an additional constraint tosolve for M as well as n simultaneously, we can enforce that

$\begin{matrix}{\frac{n^{2}}{{n^{2}M} + 1 - M} = {{( \frac{n_{L}^{2}}{{n_{L}^{2}M_{L}} + 1 - M_{L}} )v_{L}} + {( \frac{n_{P}^{2}}{{n_{P}^{2}M_{P}} + 1 - M_{P}} )v_{P}}}} & (238) \\{\frac{1}{{n^{2}M} + 1 - M} = {{( \frac{1}{{n_{L}^{2}M_{L}} + 1 - M_{L}} )v_{L}} + {( \frac{1}{{n_{P}^{2}M_{P}} + 1 - M_{P}} )v_{P}}}} & (239)\end{matrix}$

which can be subtracted to again obtain Eq. 234. There are now twoequations in two unknowns {n, M}. Expanding each term to first order inn by a Taylor expansion about n=1 and then using the fact thatv_(L)+v_(P)=1, we find that Eqs. 238-239 become respectively

[−M+M _(L) v _(L) +M _(P) v _(P)]+{−n(1−M)+(1−M _(L))n _(L) v _(L)+(1−M_(P))n _(P) v _(P)}=0  (240)

[−M+M _(L) v _(L) +M _(P) v _(P)]+{Mn−M _(L) n _(L) v _(L) −M _(P) n_(P) v _(P)}=0  (241)

where the items in square brackets are set to zero to establish theaverage value of M in a way that is consistent with the intuitive senseof the average for M. Setting the curly bracket quantities to zero alsoresults in the following expressions

$\begin{matrix}{n_{1} = {{( \frac{S_{L}}{S} )n_{L}v_{L}} + {( \frac{S_{P}}{S} )n_{P}v_{P}}}} & (242) \\{n_{2} = {{( \frac{M_{L}}{M} )n_{L}v_{L}} + {( \frac{M_{P}}{M} )n_{P}v_{P}}}} & (243) \\{M = {{M_{L}v_{L}} + {M_{P}v_{P}}}} & (244) \\{S = {{S_{L}v_{L}} + {S_{P}v_{P}}}} & (245) \\{{{\mu_{L}v_{L}} + {\mu_{P}v_{P}}} = 1} & (246) \\{{{\sigma_{L}v_{L}} + {\sigma_{P}v_{P}}} = 1} & (247) \\{{v_{L} + v_{P}} = 1} & (248) \\{{S = {1 - M}},{S_{L} = {1 - M_{L}}},{S_{P} = {1 - M_{P}}}} & (249) \\{{\mu_{L} = \frac{M_{L}}{M}},{\mu_{P} = \frac{M_{P}}{M}}} & (250) \\{{\sigma_{L} = \frac{S_{L}}{S}},{\sigma_{P} = \frac{S_{P}}{S}}} & (251) \\{{v_{L} = \frac{V_{L}}{V}},{v_{P} = \frac{V_{P}}{V}}} & (252)\end{matrix}$

where n₁ and n₂ effectively now is taken to correspond to two solutionsof a quadratic equation in n². Notice that when all particles arespherical with depolarizations M=M^(L)=M^(P)=1/3 then the equations forthe RI reduce to that of Eq. 229 as would be expected. Also, as istypical in physical analysis we usually reject one of the quadraticroots as being unphysical. In this case Eq. 243 is not a physicallyjustifiable approximation to the solution of the exact equation Eq. 234.This is because its curvature and values with respect to independentvariable v_(P) are inconsistent with the numerical solution to Eq. 234,i.e. when taking M=M_(L)v_(L)+M_(P)v_(P). It will be shown later thatthe RI is maximum when the light passes through the greatest amount ofmaterial of the particle. The rejected solution is the case that the RIis greatest when the light passes through the thinest amount of particlematerial, which is not true and is rejected for that reason. However,for the purposes of this disclosure both solutions are generallyretained as possible solutions in certain situations.

Note that the linear model given above can overestimate RI by as much as12% at the worst case volume fractions of v_(P)=50% and overestimate RIby about 3% or less at more typical volume fractions of v_(P)=12% orless. For precision optics numerical solutions of Eq. 234 should be usedor at least a second order Taylor expansion. However to understandqualitatively the underlying physics this linear model for n is quitegood. Note that Eq. 245 is derived by subtracting Eq. 244 from Eq. 248to obtain an expression for 1−M. The quantities M, which vary between0≤M≤1 are called depolarizations factors, the quantities S which varybetween 0≤S≤1 are called the shape factors. Thus, shape factors giverise to shape fractions such as up which are analogous to volumefractions up. Again, by way of review the electrical depolarizationfactors MP in the eigen symmetry basis are given by Eqs. 172-173 forsymmetrical prolate and oblate spheroids.

Using the definitions just provided we can write each diagonal elementof the RI in the eigen symmetry basis of the particles, therefore fromEq. 242

n=n _(L)σ_(L) v _(L) +n _(P)σ_(L) v _(P)  (253)

σ_(L) v _(L)+σ_(P) v _(P)=1  (254)

so that each diagonal element of the RI in the eigen symmetry basis of aparticle takes the form

n=n _(L)+(n _(P) −n _(L))v _(P)σ_(P).  (255)

This equation establishes that volume fraction and shape fraction arevery closely related and it is possible to trade off particle volumefraction v_(P) with particle shape fraction up. Said another way,particles need not be solid. It is completely possible to have adeformable particle that changes its shape by the electrokinetics ofdielectrophoresis to scatter light. Thus, by at least one of particletranslation, rotation, orientation, and material deformation light canbe scattered from one or more particles to affect a particular opticaleffect.

Going forward in this document the addition of primes to indicate theeigen symmetry basis is again reestablished. Therefore, in the eigensymmetry basis we have that the RI of particles of different shape andisotropic bulk material is given to first order as

$\begin{matrix}{\begin{bmatrix}n_{x^{\prime}y^{\prime}} & 0 & 0 \\0 & n_{y^{\prime}y^{\prime}} & 0 \\0 & 0 & n_{z^{\prime}z^{\prime}}\end{bmatrix} = {{n_{L}\overset{\_}{I}} + {( {n_{P} - n_{L}} ){v_{P}\begin{bmatrix}\sigma_{x^{\prime}x^{\prime}} & 0 & 0 \\0 & \sigma_{y^{\prime}y^{\prime}} & 0 \\0 & 0 & \sigma_{z^{\prime}z^{\prime}}\end{bmatrix}}}}} & (256)\end{matrix}$

or equivalently

n′=n _(L) I (n _(P) −n _(L))v _(P) σ _(P)′.  (257)

Equations 242-252 can be rewritten in tensor form. Start by choosing theshape of the NPs, perhaps using Eqs. 172-173 or FIG. 5 for symmetricspheroids. Then form the depolarizations matrices for components of thecolloid. For a liquid

$\begin{matrix}{\overset{=}{M_{L}^{\prime}} = \begin{bmatrix}M_{L,{x^{\prime}x^{\prime}}} & 0 & 0 \\0 & M_{L,{y^{\prime}y^{\prime}}} & 0 \\0 & 0 & M_{L,{z^{\prime}z^{\prime}}}\end{bmatrix}} & (258)\end{matrix}$

and for one type of particle it is

$\begin{matrix}{\overset{=}{M_{P}^{\prime}} = {\begin{bmatrix}M_{P,{x^{\prime}x^{\prime}}} & 0 & 0 \\0 & M_{P,{y^{\prime}y^{\prime}}} & 0 \\0 & 0 & M_{P,{z^{\prime}z^{\prime}}}\end{bmatrix}.}} & (259)\end{matrix}$

Next, determine the average tensor depolarization of the colloid interms of the volume fractions of the liquid and particles, so that

M′=M _(L) ′v _(L) +M _(P) ′v _(P)  (260)

where v_(L)+v_(P)=1. Now calculate the shape factors

S′=I−M′  (261)

S _(L)= I − M _(L) ′  (262)

S _(P) ′= I − M _(P) ′  (262)

and then form the shape fractions

σ _(L) ′=S _(L)′( S ′)⁻¹  (264)

σ _(P) ′=S _(L)′( S ′)⁻¹  (265)

where

σ _(L) ′v _(L)+σ _(P) ′v _(P) =I   (266)

v _(L) +v _(P)=1.  (267)

It should be obvious that these equations can be extended to as manyliquid and particle components as are needed to address a complexmulti-component mixture. For example, v₁+v₂+v₃=1, etc.

Finally, the RI can be calculated. In general if the bulk material ofthe NPs is crystalline, i.e. not isotropic, and the liquid of thecolloid is assumed to be isotropic, then we have

n ′=n_(L) I +( n _(P)′− n _(L) I)v _(P) σ _(P)′  (268)

where n _(P)′ is the tensor RI of the crystal in the particle-shapeeigen symmetry basis and is typically not a diagonal matrix even in theparticle's eigen symmetry basis. Also, if the NPs in a differentialvolume of a colloid are all exactly in the same orientation and “feel”the same torques then the reorientation process is describedapproximately to first order by

$\begin{matrix}\begin{matrix}{\overset{=}{n} = {\overset{=}{W}{\overset{=}{n}}^{\prime}{\overset{=}{W}}^{- 1}}} \\{= {{n_{L}\overset{=}{I}} + {{\overset{=}{W}\lbrack {( {{\overset{=}{n}}_{P}^{\prime} - {n_{L}\overset{=}{I}}} )v_{P}{\overset{=}{\sigma}}_{P}^{\prime}} \rbrack}{\overset{=}{W}}^{- 1}}}} \\{= {{n_{L}\overset{=}{I}} + {{\overset{=}{W}\lbrack {( {{\overset{=}{n}}_{P}^{\prime} - {n_{L}\overset{=}{I}}} )\overset{=}{W}\overset{=}{W}( {\overset{=}{\sigma}}^{\prime} )_{P}} \rbrack}{\overset{=}{W}}^{- 1}v_{P}}}} \\{= {{n_{L}\overset{=}{I}} + {\lbrack {{\overset{=}{W}{\overset{=}{n}}_{P}^{\prime}\overset{=}{W}} - {n_{L}\overset{=}{I}}} \rbrack \overset{=}{W}{\overset{=}{\sigma}}_{P}^{\prime}{\overset{=}{W}}^{- 1}v_{P}}}} \\{= {{n_{L}\overset{=}{I}} + {\lbrack {{\overset{=}{n}}_{P} - {n_{L}\overset{=}{I}}} \rbrack {\overset{=}{\sigma}}_{P}v_{P}}}}\end{matrix} & (269)\end{matrix}$

However, when the NPs are in random directions to start and areinfluenced by different electric fields from point-to-point then aprobability density function must be used to calculate either the mean(n) or the mean rotation matrix.

W

.Therefore,

n

≈n_(L) I +[ n _(P) −n _(L) I ]σ _(P) v _(P)  (270)

n _(P) =ξ n _(P)′ξ ⁻¹  (271)

σ _(P)=ξ σ _(P)′ξ ⁻¹  (272)

ξ=∫∫ W (θ,ϕ)ƒ_(T)(r,t,θ,ϕ,ω _(j))dθdϕ  (273)

where ξ=ξ(r, t, ω_(j)). Also, the probability density function due totorques T on NPs is ƒ_(T)(r, t, θ, ϕ, ω_(j)), which is clearly afunction of position r, time t, and the DEP excitation frequenciesω_(i). See Eq. 8 in the introduction.

Numerical Examples of Geometric Refractive Index

The previous section introduced many concepts that may become more clearby means of numerical examples. In particular, it is instructive tocalculate the tensor RI of a colloid under different conditions ofparticle geometry. Using Eqs. 242-252 for each term of the diagonal ofthe RI tensor is very instructive. Therefore, let's assume that:

1. The volume fraction of the NPs is initially v_(P)=0.10;

2. RI of a liquid silicone oil is n_(L)=1.4;

3. RI of titanium dioxide TiO₂ particles is n_(p)=2.8;

4. Liquid and particles are isotropic;

5. Liquid molecules are approximately spherical;

6. Prolate spheroids are 10× longer than diameter (e.g. Rutile TiO₂);and

7. Oblate spheroids are 10× thinner than diameter.

The volume fraction of the liquid is then v_(L)=1−v_(P)=9/10. Thediagonal elements of the depolarization tensor for the ideal sphere,prolate spheroid, and oblate spheroid are read directly from the graphof FIG. 4, whereby for spherical particles of the liquid thedepolarization tensor is

$\begin{matrix}{{\overset{=}{M_{L}^{\prime}}({Sphere})} = {\begin{pmatrix}{1/3} & 0 & 0 \\0 & {1/3} & 0 \\0 & 0 & {1/3}\end{pmatrix}.}} & (274)\end{matrix}$

The depolarization tensor for ideal prolate spheroidal particles is

$\begin{matrix}{{\overset{=}{M_{P}^{\prime}}({Prolate})} = {\begin{pmatrix}{1/2} & 0 & 0 \\0 & {1/2} & 0 \\0 & 0 & 0\end{pmatrix}.}} & (275)\end{matrix}$

The depolarization tensor for ideal oblate spheroidal particles is

$\begin{matrix}{{\overset{=}{M_{P}^{\prime}}({Oblate})} = {\begin{pmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & 1\end{pmatrix}.}} & (276)\end{matrix}$

Additionally, the colloid has an average depolarization tensor of

$\begin{matrix}{{\overset{=}{M^{\prime}}({Prolate})} = {{{v_{L}{\overset{=}{M^{\prime}}}_{L}} + {v_{P}{\overset{=}{M_{P}^{\prime}}({Prolate})}}} \approx {\begin{pmatrix}0.35 & 0 & 0 \\0 & 0.35 & 0 \\0 & 0 & 0.30\end{pmatrix}.}}} & (277)\end{matrix}$

The effective shape factor for the colloid particles is

$\begin{matrix}{{\overset{=}{S^{\prime}}({Prolate})} = {{\overset{=}{I} - {\overset{=}{M^{\prime}}({Prolate})}} \approx {\begin{pmatrix}0.65 & 0 & 0 \\0 & 0.65 & 0 \\0 & 0 & 0.70\end{pmatrix}.}}} & (278)\end{matrix}$

The shape factor for prolate spheroidal particles is

$\begin{matrix}{{\overset{=}{S_{P}^{\prime}}({Prolate})} = {{\overset{=}{I} - {\overset{=}{M_{P}^{\prime}}({Prolate})}} = {\begin{pmatrix}{1/2} & 0 & 0 \\0 & {1/2} & 0 \\0 & 0 & 1\end{pmatrix}.}}} & (279)\end{matrix}$

Therefore, the shape fraction for the prolate spheroids is

$\begin{matrix}{{\overset{=}{\sigma_{P}^{\prime}}({Prolate})} = {{{\overset{=}{S_{P}^{\prime}}({Prolate})}\lbrack {\overset{=}{S^{\prime}}({Prolate})} \rbrack}^{- 1} \approx {\begin{pmatrix}0.77 & 0 & 0 \\0 & 0.77 & 0 \\0 & 0 & 1.43\end{pmatrix}.}}} & (280)\end{matrix}$

So that, finally, for the prolate spheroid the colloid having NP volumefraction of

$\begin{matrix}{{{\overset{=}{n}}^{\prime}({Prolate})} = {{{n_{L}\overset{=}{I}} + {( {n_{P} - n_{L}} )v_{P}{{\overset{=}{\sigma}}_{P}^{\prime}({Prolate})}}} \approx {\begin{pmatrix}1.51 & 0 & 0 \\0 & 1.51 & 0 \\0 & 0 & 1.60\end{pmatrix}.}}} & (281)\end{matrix}$

And a similar set of calculations for the oblate spheroid provides

$\begin{matrix}{{{\overset{=}{n}}^{\prime}({Oblate})} = {{{n_{L}\overset{=}{I}} + {( {n_{P} - n_{L}} )v_{P}{{\overset{=}{\sigma}}_{P}^{\prime}({Oblate})}}} \approx {\begin{pmatrix}1.78 & 0 & 0 \\0 & 1.78 & 0 \\0 & 0 & 1.40\end{pmatrix}.}}} & (282)\end{matrix}$

The values in Eqs. 280-282 provide significant physical insight intowhat is occurring. The basic RI rule for prolate spheroids in a colloidis that the colloid has its maximum RI in the direction of the longestaxis of the NPs in each differential volume. The basic RI rule foroblate spheroids in a colloid is that the colloid has its minimum RI inthe direction of the shortest axis of the NPs in each differentialvolume. So the direction of propagation where the RI is greatest occurswhen the light has to pass through the greatest length of material ofthe NP, assuming the NP has a uniform RI that is greater then thesurrounding liquid medium. In Eq. 282 the oblate spheroid's thinestdirection induces no RI and the background RI of the liquid is all thatremains corresponding to the boxed value. It turns out that the oblatespheroid can provide a full range of RI values, i.e. consistent with itsvolume fraction, just by orientation of the NP. This is a previouslyunrealized capability. Had the rejected solution of Eq. 243 been usedthen Eq. 282 would have been

$\begin{matrix}{{{\overset{=}{n}}^{\prime}({Oblate})} = {{{n_{L}\overset{=}{I}} + {( {n_{P} - n_{L}} )v_{P}{{\overset{=}{\sigma}}_{P}^{\prime}({Oblate})}}} \approx {\begin{pmatrix}1.40 & 0 & 0 \\0 & 1.40 & 0 \\0 & 0 & 2.0\end{pmatrix}.}}} & (283)\end{matrix}$

The oblate sphereroid associated with Eq. 282 and Eq. 283 has a shortthickness in the z′ direction and a 10× wider diameter. So light passingthough the thickness of the oblate spheroid has much less of anopportunity to be slowed down in the higher RI of the particle. Thus theRI must be smaller in the z′ direction. Clearly this only occurs in Eq.282.

The Refractive Index Ellipsoid

An anisotropic crystal has different RI values along different principleaxes because the crystal structure provides different tendencies for theelectrons to extend to different distances from the atoms in thoseprinciple directions. This leads to different dipole moments, electricsusceptibilities and RI values in the principle directions. This isdifferent from what happens for NPs derived from isotropicmaterials andformed into non-spherical spheroids. The average RI of the colloid isdue to a time delay induced by the particles, therefore the more NPmaterial that a ray of light has to pass through the slower the averagelight propagation—i.e. when the RI of the particle is greater than theliquid.

Moreover, the total energy stored in the electric field and magneticfield of the light field is conserved. ThereforeW=W_(E)+W_(B)=2W_(E)=const. Therefore, W=E·D=D(∈₀ ∈)⁻¹D, so that in theeigen symmetry basis of a NP made from an isotropic material we have forthe colloid

$\begin{matrix}{{\frac{D_{x}^{2}/( {\epsilon_{0}W} )}{n_{x^{\prime}x^{\prime}}^{2}} + \frac{D_{y}^{2}/( {\epsilon_{0}W} )}{n_{y^{\prime}y^{\prime}}^{2}} + \frac{D_{z}^{2}/( {\epsilon_{0}W} )}{n_{z^{\prime}z^{\prime}}^{2}}} = 1} & (284)\end{matrix}$

and convert to normalized coordinates

$\begin{matrix}{{\frac{s_{x}^{2}}{n_{x^{\prime}x^{\prime}}^{2}} + \frac{s_{y}^{2}}{n_{y^{\prime}y^{\prime}}^{2}} + \frac{s_{z}^{2}}{n_{z^{\prime}z^{\prime}}^{2}}} = 1} & (285)\end{matrix}$

and therefore take s_(x)=s_(y)=n(θ) sin θ and s_(z)=n(θ) cos θ, where θis the angle from the optical axis, i.e. the z′ axis. Therefore, forboth prolate and oblate spheroids n_(x′x′)=n_(y′y′)=n_(o)=n_(a) is theordinary RI, and n_(z′z′)=n_(e)=n_(c) is the extraordinary RI. Thedesignations n_(a) and n_(c) help connect RI to geometry as theyindicate the RI along the a and c axis respectively, where we note thata=b for symmetric prolate and oblate spheroids.

Hence, the RI for a colloid with all the NPs having the same 0 in adifferential volume is given by

$\begin{matrix}{\frac{1}{n^{2}(\theta)} = {\frac{\sin^{2}\theta}{n_{a}^{2}} + \frac{\cos^{2}\theta}{n_{c}^{2}}}} & (286)\end{matrix}$

where n_(a)<n_(c) for prolate spheroids, n_(a)>n_(c) for oblatespheroids, and n_(a)=n_(c) for spheres. This can be extended to NPs withan underlying crystal anisotropy and geometric anisotropy. Thus, acolloid comprising NPs with non-spherical geometry exhibit differenteffective RI values in different directions, even though the underlyingmaterial that comprises the NPs is isotropic and homogeneous in theparticles. Thus, by varying the volume fraction of particles v_(P) aswell as the orientation of particles σ _(P)′ the effective RI of

$\begin{matrix}{{\overset{=}{n}}^{\prime} = {\begin{bmatrix}n_{a} & 0 & 0 \\0 & n_{a} & 0 \\0 & 0 & n_{c}\end{bmatrix} = {{n_{L}\overset{=}{I}} + {( {n_{P} - n_{L}} )v_{P}{\overset{=}{\sigma}}_{P}^{\prime}}}}} & (287)\end{matrix}$

manifests itself not just in the effective RI of a NP, but in theeffective RI of the colloid in a differential volume. Obviously, if allthe particles are in random directions then an averaging process overall NPs needs to be undertaken to assess the effective RI as a single NPhas insufficient information.

Assuming that n_(P)>n_(L) then a wave moving along a trajectory havingmore high RI material will experience a slower average speed and ahigher effective RI. For prolate spheroids, a lightwave moving along thec axis of many particles experiences a greater delay and a highereffective RI. For oblate spheroids, a lightwave moving along the a=baxis (or more generally radial axis) of many NPs experiences a greaterdelay and a higher effective RI. Thus, Eqs. 286-287 provides a usefulway to calculate the RI in a particular direction and is of greatutility to the optical scientist.

Forces and Torques on Particles

This section will focus only on the general equations for forces andtorques that may be developed by an external field on a particle. Ingeneral, these forces and torques may be applied to isotropic,anisotropic, electrically lossless and electrically lossy particles.However, the effects for each type of particle can be different.

Time-Domain Force Equation

Let the position vector r point form the origin of the observer's frameof reference to a point that is midway between the effective positiveand negative point charges forming a dipole, then the resulting force ona neutral NP is

$\begin{matrix}{{\mathcal{F}( {r,t} )} = {{{q( {t - \tau} )}{ɛ_{in}\lbrack {{r + \frac{d( {t - \tau} )}{2}},t} \rbrack}} + {\{ {- {q( {t - \tau} )}} \} {ɛ_{in}\lbrack {{r - \frac{d( {t - \tau} )}{2}},t} \rbrack}}}} & (288)\end{matrix}$

where τ is a delay that characterizes the amount of time it takes or thecharges to respond to the input field ξ_(in)(r, t). The expression abovemay be spatially Taylor expanded to 1^(st) order so that

$\begin{matrix}{{\mathcal{F}( {r,t} )} = {{{q( {t - \tau} )}\{ {{ɛ_{in}( {r,t} )} + {{\nabla{ɛ_{in}( {r,t} )}} \cdot \frac{d( {t - \tau} )}{2}}} \}} - {{q( {t - \tau} )}\{ {{ɛ_{in}( {r,t} )} = {{\nabla{ɛ_{in}( {r,t} )}} \cdot \frac{d( {t - \tau} )}{2}}} \}}}} & (289)\end{matrix}$

and therefore

(r, t)=q(t−τ)d(t−τ)·∇ξ_(in)(r, t). The time delayed and dipole moment atr is p_(par)(t−τ)=q(t−τ)d(t−ξ). Also, to the extent that polarization ofparticles occurs at different r for different NPs in a colloid the forceon a singe NP is then a function of position whereby

(r,t)=p _(par)(r,t−τ)·∇ξ_(in)(r,t).  (290)

Time-Domain Torque Equation

The resulting torque on a similarly disposed neutral NP is given interms of the vector cross product x, so that

$\begin{matrix}{{( {r,t} )} = {{\frac{d( {t - \tau} )}{2} \times {q( {t - \tau} )}{ɛ_{in}\lbrack {{r + \frac{d( {t - \tau} )}{2}},t} \rbrack}} + {\frac{- {d( {t - \tau} )}}{2} \times \{ {- {q( {t - \tau} )}} \} {ɛ_{in}\lbrack {{r + \frac{d( {t - \tau} )}{2}},t} \rbrack}}}} & (291)\end{matrix}$

which on Taylor expansion, i.e. in a way very similar to that shownabove for force, provides the torque as

(r,t)=p _(par)(T,t−τ)×ξ_(in)(r,t).  (292)

Properties of the Time-Domain Equations

Based on the last two sections we see that the time-domain masterequations for force and torque on a NP are

(r,t)=p _(par)(r,t−τ)·∇ξ_(in)(r,t)  (293)

(r,t)=p _(par)(r,t−τ)×∇ξ_(in)(r,t)  (294)

These equations tell two important things. First, that a force-basedtranslation of a NP within a fluid requires a non-uniform electricfield; second, that torque based orientation of a NP requires ananisotropic NP structure, which ensures that the dipole moment is not inthe same direction as the input field ξ_(in), otherwise the torque iszero. The anisotropic structure of the NP for orientation may come fromNP shape, NP material composition, or both. Thus, a uniform electricfield can rotate NPs, but it cannot translate the NP. A non-uniformelectric field can both translate and orient a NP. Finally note thatparticle deformation of soft particles (e.g. emulsions) across asignificant range of spheroidal shapes (e.g. from prolate to oblatespheroidal) requires an asymmetric force across the particles toinitially change its shape from spherical and then sheer stresses due totorques to change its orientation. This requires a combination ofapplied fields. Other modes of deformation are possible. To make furtherprogress, a general expression for p(t−τ) for typical particles ofinterest is developed in the next two sub-sections.

Frequency-Domain Force Equation

From Eq. 69 we know that there is a direct connection between the dipolemoment and the input electric field. Therefore, the induced dipolemoment can be written as

p _(par)(r,t−τ)=∈_(med)∇|χ(ω)|ξ_(in)(r,t−τ(ω))  (295)

where ∈_(med) is the dielectric constant of the medium surrounding theparticle and ν is the volume of the particle. The induced time lag ist=ξ(ω) and the amplitude scaling is |χ|=|χ(ω)|. Therefore, from Eq. 293

(r,t)=∈_(med) νV|χ(ω)|ξ_(in)(r,t−τ(ω))·∇ξ_(in)(r,t).  (296)

However, in each coordinate direction σ∈{x, y, z} a harmonic electricfield has components

ξ_(in,σ)(r,t)=E _(in,σ)(r)cos[ωt−ϕ _(σ)]=cos ωt[E _(σ)cos ϕ_(σ)]+sinωt[E _(σ)sin ϕ_(σ)]  (297)

where E_(ν)=E_(σ)(r, ω) and ϕ_(σ)=ϕ_(σ)(r, ω). The components in squarebrackets are phasor components of E_(a), therefore

ξ_(in)(r,t)=cos ωtE _(R)+sin ωt E _(I)  (298)

where E_(R)=

E_(x) cos ϕ_(x), E_(y) cos ϕ_(y), E_(z) cos ϕ_(z)

and similarly it is found that E_(I)=

(E_(z) sin ϕ_(x), E_(y) sin ϕ_(y), E_(z) sin ϕ_(z)

. Therefore, by using Eq. 298

ξ_(in)(r,t−τ)=cos ωt[E _(R) cos ωτ−E _(I) sin ωτ]+sin ωt[E _(R) sin ωτ+E_(I) cos ωτ]   (299)

and if we identify ψ(ω)=ωτ(ω) as the time-shift induced-phase then thereal and imaginary parts of the electric susceptibility areω_(R)=|K(ω)|cos ψ(ω) and ω_(I)=|ω(ω)sin ψ(ω). Therefore

|χ|ξ_(in)(r,t−τ)=[E _(R)χ_(R) −E _(I)χ_(I)] cos ωt+[E _(R)χ_(I) +E_(I)χ_(R)]sin ωt.  (300)

By utilizing Eq. 298 and 300 in Eq. 296 the instantaneous force on a NPbecomes

$\begin{matrix}{\frac{\mathcal{F}( {r,t} )}{\epsilon_{med}V} = {| \chi \middle| {{ɛ_{in}( {r,{t - \tau}} )} \cdot {\nabla{ɛ_{in}( {r,t} )}}}  = {{\cos^{2}{{\omega t}\lbrack {{\chi_{R}{E_{R} \cdot {\nabla E_{R}}}} - {\chi_{I}{E_{I} \cdot {\nabla E_{R}}}}} \rbrack}} + {\sin^{2}{{\omega t}\lbrack {{\chi_{I}{E_{R} \cdot {\nabla E_{I}}}} + {\chi_{R}{E_{I} \cdot {\nabla E_{I}}}}} \rbrack}} + {{{sin\omega}t{cos\omega}t}\lbrack {{''}{{Don}'}t\mspace{14mu} {{Care}{''}}} \rbrack}}}} & (301)\end{matrix}$

However, the average force is more useful in determining the effects ofparticle motion. The time average

cos² ωt

=

sin² ωt

=1/2, however the time average

sin ωt sin ωt

=0. This explains why “Don't Care” is in the third term in Eq. 301.Therefore, we can write

$\begin{matrix}{F = {\frac{\epsilon_{med}V}{2}\lbrack \{ {{( {\chi_{R}E_{R}} ) \cdot {\nabla E_{R}}} +  \quad{( {\chi_{R}E_{I}} ) \cdot {\nabla E_{I}}} \} + \{ {{( {\chi_{I}E_{R}} ) \cdot {\nabla E_{I}}} - {( {\chi_{I}E_{I}} ) \cdot {\nabla E_{R}}}} \}} \rbrack }} & (302)\end{matrix}$

However, if this force is along a eigen symmetry basis of the electricsusceptibility then F is just the force F₁ along the first eigen basisdirection and the complex scalar χ=χ_(R)−iχ_(I) is just the firstdiagonal element in a complex-valued diagonal susceptibility tensor χ.Thus, by the principle of superposition of forces it is possible to addtogether the separate forces F=F₁+F₂+F₃ (and associated equations) dueto each of the eigen symmetry directions of the electric susceptibility.Each equation is associated with a different diagonal position in thesusceptibility tensor. This can subsequently be converted to any desiredbasis other than the eigen symmetry basis of the electricsusceptibility. The general result is

$\begin{matrix}{F = {\frac{\epsilon_{med}V}{2}\lbrack \{ {{( {{\overset{=}{\chi}}_{R}E_{R}} ) \cdot {\nabla E_{R}}} +  \quad{( {{\overset{=}{\chi}}_{R}E_{I}} ) \cdot {\nabla E_{I}}} \} + \{ {{( {{\overset{=}{\chi}}_{I}E_{R}} ) \cdot {\nabla E_{I}}} - {( {{\overset{=}{\chi}}_{I}E_{I}} ) \cdot {\nabla E_{R}}}} \}} \rbrack }} & (303) \\{\mspace{79mu} {where}} & \; \\{\mspace{79mu} {E_{in} = {E_{R} + {iE}_{I}}}} & (304) \\{\mspace{79mu} {\overset{=}{\chi} = {{\overset{=}{\chi}}_{R} + {i{\overset{=}{\chi}}_{I}}}}} & (305)\end{matrix}$

Note that E_(R) and E_(I) need not be in the same direction. Of specialinterest is when they are orthogonal to each other so there is acirculating field. Moreover, in the interest of a compact expression forthe force equation it should be noted that by using Eqs. 304 and 305 itis easy to rewrite Eq. 303 as

$\begin{matrix}{F = {\frac{\epsilon_{med}V}{2}{{Re}\lbrack {\overset{=}{\chi}{E_{in} \cdot {\nabla E_{in}^{*}}}} \rbrack}}} & (306)\end{matrix}$

where Re[·] represents the real part of the expression in brackets andE_(in)* is the complex conjugate of the electric field intensity.

In the specific case when the susceptibility is a scalar (or if in itsdiagonal eigen symmetry basis) then in Eq. 302 we can identify|E_(in)|²=E_(R)·E_(R)+E_(I)·E_(I) and noting the formE_(in)·∇E_(in)=∇|E_(in)|²/2 we obtain

$\begin{matrix}{{\langle\frac{\mathcal{F}( {r,t} )}{\epsilon_{med}V}\rangle} = {\frac{1}{2}\lbrack {\frac{\chi_{R}}{2}\nabla} \middle| E \middle| {}_{2}{+ {\chi_{I}( {{E_{R} \cdot {\nabla E_{I}}} - {E_{I} \cdot {\nabla E_{R}}}} )}} \rbrack}} & (307)\end{matrix}$

However, we can use the identity∇×(E_(I)×E_(R))=E_(R)·∇E_(I)·E_(I)·∇E_(R) for solenoidal fields∇·E_(R)=∇·E_(I)=0, which is the case when no free charges exist within aparticle for both a good conductor and a good insulator. The result isthe average quasielectrostatic pondermotive force F=

on a particle:

$\begin{matrix}{F = {{\frac{\epsilon_{med}V}{2}\lbrack {\frac{\chi_{R}}{2}\nabla} \middle| E_{in} \middle| {}_{2}{{+ \chi_{I}}{\nabla{\times ( {E_{I} \times E_{R}} )}}} \rbrack}.}} & (308)\end{matrix}$

In the eigen symmetry basis of the electric susceptibility we have thenthat

$\begin{matrix}{F^{\prime} = {{\frac{\epsilon_{med}V}{2}\lbrack {\frac{\chi_{R}^{\prime}}{2}\nabla} \middle| E_{in} \middle| {}_{2}{{+ \chi_{I}^{\prime}}{\nabla{\times ( {E_{I}^{\prime} \times E_{R}^{\prime}} )}}} \rbrack}.}} & (309)\end{matrix}$

which can be converted to the observer's frame of reference by F=WF′W⁻¹. Finally, as was noted earlier, the details of the physicalparameters of size, shape, and composition are encapsulated in χ.

Frequency-Domain Torque Equation

It is possible to leverage the development of the previous section.Therefore, consider Eq. 292 and evaluate it using Eqs 295, 298, 300,304, and 305 so that the instantaneous torque on a NP becomes

$\begin{matrix}{\frac{( {r,t} )}{\epsilon_{med}V} = {{\overset{=}{\chi}{ɛ_{in}( {r,{t - \tau}} )} \times {ɛ_{in}( {r,t} )}} = {{\cos^{2}{{\omega t}\lbrack {{( {{\overset{=}{\chi}}_{R}E_{R}} ) \times E_{R}} - {( {{\overset{=}{\chi}}_{I}E_{I}} ) \times E_{R}}} \rbrack}} + {\sin^{2}{{\omega t}\lbrack {{( {{\overset{=}{\chi}}_{I}E_{R}} ) \times E_{I}} + {( {{\overset{=}{\chi}}_{R}E_{I}} ) \times E_{I}}} \rbrack}} + {{{sin\omega}t{cos\omega}t}\lbrack {``{{{Don}'}t\mspace{14mu} {Care}}"} \rbrack}}}} & (310)\end{matrix}$

which is similar to Eq. 301 for force, but with the dot product into thegradient operator replaced with a vector cross product. Next, we canagain exploit the time averages

cos² ωt

=

sin² ωt

=1/2 and

sin ωt sin ωt

=0. Therefore, the time average torque (

) is

$\begin{matrix}{T = {\frac{\epsilon_{med}V}{2}\lbrack {\{ {{( {{\overset{=}{\chi}}_{R}E_{R}} ) \times E_{R}} + {( {{\overset{=}{\chi}}_{R}E_{I}} ) \times E_{I}}} \} + \{ {{( {{\overset{=}{\chi}}_{I}E_{R}} ) \times E_{I}} - {( {{\overset{=}{\chi}}_{I}E_{I}} ) \times E_{R}}} \}} \rbrack}} & (311)\end{matrix}$

Note that [χ _(R)E_(R)]×E_(R)≠χ _(R)[E_(R)×E_(R)]=0. Moreover, by usingthe definitions E_(in)=E_(R)+iE_(I) and χ=χ _(R)−iχ_(I) then Eq. 311 canbe rewritten as

$\begin{matrix}{T = {\frac{\epsilon_{med}V}{2}{{Re}\lbrack {\overset{=}{\chi}E_{in} \times E_{in}^{*}} \rbrack}}} & (312)\end{matrix}$

where Re[·] represents the real part of the expression in brackets andE* is the complex conjugate of the electric field intensity. Note thatthe real and imaginary parts of the electric field E need not be in thesame direction. Therefore, it is possible for the susceptibility to be ascalar as in χ=Iχ and still have a non-zero torque.

Electrorotation Using Quadrature Electric Fields

In this subsection, we consider the case of exciting a particle withquadrature electric fields, i.e. orthogonal fields that are π/2 radiansout of phase. Consider an arbitrary shaped particle. Then from Eq. 69and Eq. 312 and we have

T=½Re[p _(par) ×E _(in)*]  (313)

p _(par)=∈_(med) νχE _(in)  (314)

however, in terms of the real and imaginary parts of the electricsusceptibility in the eigen symmetry basis

χ′=χ _(R) ′+iχ _(I)′.  (315)

Let's assume that the particle has its center of mass at the origin sothat a rotating electric field in the eigen symmetry basis is given by

E=(E _(x) {circumflex over (x)}±iE _(y′) ŷ′).  (316)

The induced dipole moment in the eigen symmetry basis is then

p _(par)′=∈_(med)ν[(χ _(R) ′+iX _(I)′)·(E _(x′) x′±iE _(′) y ′)]  (317)

Therefore, we find that the torque is

$\begin{matrix}{T = {{\frac{\epsilon_{med}{VE}_{x^{\prime}}E_{y^{\prime}}}{2}\lbrack {\chi_{I,x^{\prime},x^{\prime}} \pm \chi_{I,{y^{\prime}y^{\prime}}}} \rbrack}{\overset{\sim}{z}.}}} & (318)\end{matrix}$

So by driving the NPs in quadrature from different orthogonal directionsit is possible to induce a torque and pure rotation—this is called theBorn-Lortes effect. Moreover, if the phase of the y′ component leads byπ/2 over the x′ the torque is proportional to [χ_(I,x′x′)+χ_(I,y′y′)].If the phase of the y′ component lags by π/2 over the x′ the torque isproportional to [χ_(I,x′x′)+χ_(I,y′y′)]. It is a straightforward matterto put the torque in terms of the constitutive parameters by using Eq.192. This effect allows the particle to be rotated continuously aboutthe z′-axis if E_(z′)=0. Thus, with this type of rotation it is, inprinciple, possible to make a time crystal in a fluidic meta-material,with the properties of the crystal varying in time by the rotation ofNPs. This is especially easy to see if the particles are non-spherical.Moreover, if the effect is utilized for a short time only it allowsreorientation by rotation. Rotation over short times is equivalent to areorientation process.

Frequency and Phase Selective Particle Orientation

In this section the general equations for determining a NP'sstability-axis are developed. The starting point for analyzing a NP'sstability axis is from Eq. 69 and Eq. 312, whereby

p _(par)=∈_(med) νχE _(in)  (319)

T=½Re[p _(par) ×E _(in)*].  (320)

Also, the elements of the diagonal form χ′ are obtained from Eq. 192

$\begin{matrix}{{\chi^{\prime} \approx {\frac{K_{\infty}}{1 - {{\beta M}^{\prime}K_{\infty}}} + \frac{\frac{K_{0}}{1 - {{\beta M}^{\prime}K_{0}}} - \frac{K_{\infty}}{1 - {{\beta M}^{\prime}K_{\infty}}}}{1 + {\xi^{2}\tau_{MW}^{2}\omega^{2}}} + {i\frac{\lbrack \frac{K_{\infty} - K_{0}}{( {1 - {{\beta M}^{\prime}K_{0}}} )^{2}} \rbrack \tau_{MW}\omega}{1 + {\xi^{2}\tau_{MW}^{2}\omega^{2}}}}}},} & (321)\end{matrix}$

where we take

$\begin{matrix}{\chi_{R}^{\prime} = {\frac{K_{\infty}}{1 - {{\beta M}^{\prime}K_{\infty}}} + \frac{\frac{K_{0}}{1 - {{\beta M}^{\prime}K_{0}}} - \frac{K_{\infty}}{1 - {{\beta M}^{\prime}K_{\infty}}}}{1 + {\xi^{2}\tau_{MW}^{2}\omega^{2}}}}} & (322) \\{\chi_{I}^{\prime} = \frac{\lbrack \frac{K_{\infty} - K_{0}}{( {1 - {{\beta M}^{\prime}K_{0}}} )^{2}} \rbrack \tau_{MW}\omega}{1 + {\xi^{2}\tau_{MW}^{2}\omega^{2}}}} & (323)\end{matrix}$

which comprises factors from Eqs. 193-198

$\begin{matrix}{\beta = {\frac{v_{par}}{v_{\max}} \approx {( \frac{3}{2} )v_{par}}}} & (324) \\{K_{0} = \frac{\sigma_{pR} - \sigma_{mR}}{{M^{\prime}\sigma_{pR}} + {( {1 - M^{\prime}} )\sigma_{mR}}}} & (325) \\{K_{\infty} = \frac{\epsilon_{pR} - \epsilon_{mR}}{{M^{\prime}\epsilon_{pR}} + {( {1 - M^{\prime}} )\epsilon_{mR}}}} & (326) \\{\tau_{MW} = \frac{{M^{\prime}\epsilon_{pR}} + {( {1 - M^{\prime}} )\epsilon_{mR}}}{{M^{\prime}\sigma_{pR}} + {( {1 - M^{\prime}} )\sigma_{mR}}}} & (327) \\{\xi = \frac{1 - {K_{\infty}{\beta M}^{\prime}}}{1 - {K_{0}{\beta M}^{\prime}}}} & (328) \\{{\xi\tau}_{MW} = {\frac{{\epsilon_{mR}( {1 - {M^{\prime}( {1 - \beta} )}} )} + {{M^{\prime}( {1 - \beta} )}\epsilon_{pR}}}{{\sigma_{mR}( {1 - {M^{\prime}( {1 - \beta} )}} )} + {{M^{\prime}( {1 - \beta} )}\sigma_{pR}}}.}} & (329)\end{matrix}$

For example, the eigen symmetry basis of a particle x′ is diagonal andthe x′-component of the torque can be written in terms of a generalfield

E _(in) ′=E _(x′) e ^(iϕ) ^(x) ′{circumflex over (x)}+E _(y′) e ^(iϕ)^(y) ′ŷ+E _(z′) e ^(iϕ) ^(z) ′{circumflex over (z)}  (330)

χ′={circumflex over (χ)}_(R) ′+iχ _(I)′  (331)

We then find the components of the torque in the eigen symmetry basis as

$\begin{matrix}{T_{x^{\prime}} = {\frac{\epsilon_{med}V}{2}E_{y^{\prime}}{E_{z^{\prime}}\lbrack {{( {\chi_{{Ry}^{\prime}y^{\prime}} - \chi_{{Rz}^{\prime}z^{\prime}}} ){\cos ( {\varphi_{y^{\prime}} - \varphi_{z^{\prime}}} )}} - {( {\chi_{{Iy}^{\prime}y^{\prime}} + \chi_{{Iz}^{\prime}z^{\prime}}} ){\sin ( {\varphi_{y^{\prime}} - \varphi_{z^{\prime}}} )}}} \rbrack}}} & (332) \\{T_{y^{\prime}} = {\frac{\epsilon_{med}V}{2}E_{z^{\prime}}{E_{x^{\prime}}\lbrack {{( {\chi_{{Rz}^{\prime}z^{\prime}} - \chi_{{Rx}^{\prime}x^{\prime}}} ){\cos ( {\varphi_{z^{\prime}} - \varphi_{x^{\prime}}} )}} - {( {\chi_{{Iz}^{\prime}z^{\prime}} + \chi_{{Ix}^{\prime}x^{\prime}}} ){\sin ( {\varphi_{z^{\prime}} - \varphi_{x^{\prime}}} )}}} \rbrack}}} & (333) \\{T_{z^{\prime}} = {\frac{\epsilon_{med}V}{2}E_{x^{\prime}}{E_{y^{\prime}}\lbrack {{( {\chi_{{Rx}^{\prime}x^{\prime}} - \chi_{{Ry}^{\prime}y^{\prime}}} ){\cos ( {\varphi_{x^{\prime}} - \varphi_{y^{\prime}}} )}} - {( {\chi_{{Ix}^{\prime}x^{\prime}} + \chi_{{Iy}^{\prime}y^{\prime}}} ){\sin ( {\varphi_{x^{\prime}} - \varphi_{y^{\prime}}} )}}} \rbrack}}} & (334)\end{matrix}$

However, this is a complicated set of expressions, especially whenelectric field amplitudes and phases are transformed from the observer'sframe of reference to the particle's frame of reference and theappropriate substitutions are made for the real and imaginary parts ofthe electric susceptibility in terms of the constitutive parameters.

Therefore to make some progress with analysis let us initially assume(assumption 1) that the eigen symmetry basis of the particle is the sameas the observer's frame of reference. This means that x=x′, y=y′, andz=z′. Therefore, the amplitudes and phases of the electric field do nothave to be transformed from the observer's frame of reference to theparticle's frame of reference. Additionally, let's assume (assumption

2) that the applied electric field comes only from the z-direction, thus∈_(x)=∈_(y)=0, and ∈_(z)>0. This sets all torques identically to zero.

However, we can now assume that a small perturbation to the position ofthe electric fields is introduced so that in the particle's eigensymmetry basis is not aligned to the electric fields δE_(x)>0 andδE_(y)>0 (assumption

3), which can be used to examine the stability of the particle and findthe stability axes. It is easy to see that

$\begin{matrix} {{\delta T}_{x} \approx {\frac{\epsilon_{med}V}{2}{\delta E}_{y} {E_{z}\lbrack {{( {\chi_{Ryy} - \chi_{Rzz}} ){\cos ( {\varphi_{y} - \varphi_{z}} )}} -}\quad }( {\chi_{Iyy} + \chi_{Izz}} ){\sin ( {\varphi_{y} - \varphi_{z}} )}}} \rbrack & (335) \\ {{\delta T}_{y} \approx {\frac{\epsilon_{med}V}{2}E_{z} \delta  {E_{x}\lbrack {{( {\chi_{Rzz} - \chi_{Rxx}} ){\cos ( {\varphi_{z} - \varphi_{x}} )}} -}\quad }( {\chi_{Izz} + \chi_{Ixx}} ){\sin ( {\varphi_{z} - \varphi_{x}} )}}} \rbrack & (336) \\ {{\delta T}_{z} \approx {\frac{\epsilon_{med}V}{2}{\delta E}_{x} \delta  {E_{y}\lbrack {{( {\chi_{Rxx} - \chi_{Ryy}} ){\cos ( {\varphi_{x} - \varphi_{y}} )}} -}\quad }( {\chi_{Ixx} + \chi_{Iyy}} ){\sin ( {\varphi_{x} - \varphi_{y}} )}}} \rbrack & (337)\end{matrix}$

and then assume that the phase differences are all zero (assumption

4) and then note that the second order fields can be set to zero in theusual way so that

δT _(x) ∝δE _(y) E _(z)[χ_(Ryy)−χ_(Rzz)]  (338)

δT _(y) ∝E _(z) δE _(x)[χ_(Rzz)−χ_(Rxx)]  (339)

δT _(z)∝0  (340)

Next, use Eqs. 322-323 and identify the real and imaginary parts of theelectric susceptibility (only the real part is needed in this exampleanalysis)

$\begin{matrix}{\chi_{Ryy} = {\frac{K_{\infty}}{1 - {{\beta M}_{yy}K_{{yy},\infty}}} + \frac{\frac{K_{{yy},0}}{1 - {{\beta M}_{yy}K_{{yy},0}}} - \frac{K_{{yy},\infty}}{1 - {{\beta M}_{yy}K_{{yy},\infty}}}}{1 + {\xi_{yy}^{2}\tau_{{yy},{MW}}^{2}\omega^{2}}}}} & (341) \\{\mspace{79mu} {and}} & \; \\{\chi_{Rzz} = {\frac{K_{\infty}}{1 - {{\beta M}_{zz}K_{{zz},\infty}}} + \frac{\frac{K_{{zz},0}}{1 - {{\beta M}_{zz}K_{{zz},0}}} - \frac{K_{{zz},\infty}}{1 - {{\beta M}_{zz}K_{{zz},\infty}}}}{1 + {\xi_{zz}^{2}\tau_{{zz},{MW}}^{2}\omega^{2}}}}} & (342)\end{matrix}$

and on further use of Eqs. 324-329 we obtain for the differentialtorques

$\begin{matrix}{{\delta T}_{x} \propto {{\delta E}_{y}{E_{z}\lbrack {\underset{{Term}\mspace{14mu} {A1}}{\underset{}{\frac{( {M_{zz} - M_{yy}} )( {1 - \beta} )( {\epsilon_{mR} - \epsilon_{pR}} )}{{g\lbrack {{\epsilon_{mR}.\epsilon_{pR}.M_{yy}},\beta} \rbrack}{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{zz},\beta} \rbrack}}}} + \underset{\underset{{Term}\mspace{14mu} {A2}}{}}{\frac{\frac{{\epsilon_{mR}\sigma_{pR}} - {\epsilon_{pR}\sigma_{mR}}}{{g\lbrack {{\epsilon_{mR}.\epsilon_{pR}},M_{yy},\beta} \rbrack}{g\lbrack {{\sigma {mR}},\sigma_{pR},M_{yy},\beta} \rbrack}}}{1 + {( \frac{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{yy},\beta} \rbrack}{g\lbrack {\sigma_{mR},\sigma_{pR},M_{yy},\beta} \rbrack} )^{2}\omega^{2}}}} - \underset{\underset{{Term}\mspace{14mu} {A3}}{}}{\frac{\frac{{\epsilon_{mR}\sigma_{pR}} - {\epsilon_{pR}\sigma_{mR}}}{{g\lbrack {{\epsilon_{mR}.\epsilon_{pR}},M_{zz},\beta} \rbrack}{g\lbrack {{\sigma {mR}},\sigma_{pR},M_{zz},\beta} \rbrack}}}{1 + {( \frac{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{zz},\beta} \rbrack}{g\lbrack {\sigma_{mR},\sigma_{pR},M_{zz},\beta} \rbrack} )^{2}\omega^{2}}}}} \rbrack}}} & (343)\end{matrix}$

and a similar expression for δT_(x), with appropriate exchanges ofsubscripts, is

$\begin{matrix}{{\delta T}_{y} \propto {E_{z}{{\delta E}_{x}\lbrack {\underset{{Term}\mspace{14mu} {B1}}{\underset{}{\frac{( {M_{xx} - M_{zz}} )( {1 - \beta} )( {\epsilon_{mR} - \epsilon_{pR}} )}{{g\lbrack {{\epsilon_{mR}.\epsilon_{pR}.M_{zz}},\beta} \rbrack}{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{xx},\beta} \rbrack}}}} + \underset{\underset{{Term}\mspace{14mu} {B2}}{}}{\frac{\frac{{\epsilon_{mR}\sigma_{pR}} - {\epsilon_{pR}\sigma_{mR}}}{{g\lbrack {{\epsilon_{mR}.\epsilon_{pR}},M_{zz},\beta} \rbrack}{g\lbrack {{\sigma {mR}},\sigma_{pR},M_{zz},\beta} \rbrack}}}{1 + {( \frac{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{zz},\beta} \rbrack}{g\lbrack {\sigma_{mR},\sigma_{pR},M_{zz},\beta} \rbrack} )^{2}\omega^{2}}}} - \underset{\underset{{Term}\mspace{14mu} {B3}}{}}{\frac{\frac{{\epsilon_{mR}\sigma_{pR}} - {\epsilon_{pR}\sigma_{mR}}}{{g\lbrack {{\epsilon_{mR}.\epsilon_{pR}},M_{xx},\beta} \rbrack}{g\lbrack {{\sigma {mR}},\sigma_{pR},M_{xx},\beta} \rbrack}}}{1 + {( \frac{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{xx},\beta} \rbrack}{g\lbrack {\sigma_{mR},\sigma_{pR},M_{xx},\beta} \rbrack} )^{2}\omega^{2}}}}} \rbrack}}} & (344)\end{matrix}$

where the auxiliary function g is given by

g[m,p,M,β]=pM(1−β)+m(1−M(1−β)).  (345)

where m represents the medium quantity, p represents the particlequantity, M is the associated depolarization, and β is the compactnessparameter. The resulting differential torques of Eqs. 343-344 havecertain properties, which are described next.

Let's assume that a NP is a tri-axial spheroid with the principle axis{a, b, c} along the {x′, y′, z′} axes respectively and therefore alsoalong the {x, y, z}axes respectively by assumption

1. The spheroid has principle lengths 0<a<b<c (assumption

5) so that depolarization factors are 0<M_(z′z′)<M_(y′y′)<M_(x′x′)<1.Therefore, the torque directions along each axis {x′, y′, z′} arewell-defined. Observe that the terms (A1, B1) are independent of thereal part of the conductivity of the medium and particle {σ_(mR),σ_(pR)}, but terms (A2, A3, B2, B3) are strong functions of theconductivities as can be seen by inspection according to Eqs. 343-344.

CASE 1: if the colloid is lossless so that the conductivities are zero,then only the terms (A1, B1) survive and these terms are independent ofthe DEP frequency of excitation ω. Alternately, if the conductivitiesare not zero, but the frequency of DEP excitation is large (ω→∞) thenagain only the (A1, B1) terms survive.

In the cases where only terms (A1, B1) survive we have set the initialconditions as

a initially along axis x=x′  (346)

b initially along axis y=y′  (347)

c initially along axis z=z′  (348)

0<a<b<c  (349)

0<M _(zz) <M _(yy) <M _(xx)<1  (350)

E _(x)=0,E _(y)=0,E _(z)>0  (351)

δE _(x)=0,δE _(y)=0,δE _(z)>0  (352)

δT _(x) ∝δE _(y) E _(z)(M _(zz) −M _(yy))=0  (353)

δT _(y) ∝δE _(z) δE _(x)(M _(xx) −M _(zz))=0  (354)

δT _(z)=0.  (355)

which provides a particle initially devoid of torque applied to it andwith a specific orientation. Then a small perturbation of the electricfield direction is made so that the electric field points into thatregion of space where x>0, y>0, and z>0. Then we find that

E _(x)>0,E _(y)>0,E _(z)>0  (356)

δE _(x)>0,δE _(y)>0,δE _(z)>0,  (357)

which results in

δT _(x) ∝δE _(y) E _(z)(M _(zz) −M _(yy))<0  (358)

δT _(y) ∝E _(z) δE _(z)(M _(zz) −M _(zz))<0  (359)

δT ₂∝0.  (360)

This drives the long axis of the spheroid towards the direction of theelectric field. Once the particle aligns with the electric field againthen all torques become zero and the system is stable once again. Saidanother way, the long axis of the prolate spheroid aligns with thedirection of the electric field and tends to track the direction of theelectric field. If, instead of aligning the long axis of the prolatespheroid with the electric field, a short axis of the field was alignedthen initially, the system would have no torques, but as soon as theelectric field was slightly redirected then the torques on the particlewould be such as to rotate the particle's short axis away from the fieldand again try to align the particle's long axis with the electric field.A similar analysis for an oblate spheroid (i.e. a coin-like shape with0<c<a<b) shows that the radial b-axis of an approximately coin-likeparticle aligns with the direction of the electric field. This is theopposite type of response that is provided from the nearly prolatespheroid.

In summary of case 1, when (1) the permittivity of the particle isgreater than that of the surrounding medium and the DEP frequency ishigh; or if (2) the permittivity of the particle is greater than that ofthe surrounding medium, the colloid is lossless, and at any DEPfrequency; then in both cases torques develop that tend to align thelong-axis of each particle towards the direction of the electric field.For completeness, note that the long axis of a NP is also aligned withthe electric field at ω=0.

CASE 2: if the colloid is not lossless and not necessarily at a highfrequency then all terms (A1, A2, A3, B1, B2, B3) of Eqs. 343-344survive. Then the total torques become a strong function of the DEPfrequency b and the sign of the torques can change from that discussedin case-1. This means that the stable axis is no longer restricted tothe longer axis of the NP as being the stable axis. The reason for thisis that each axis of the NP is associated with its own Maxwell-Wignertime constant and there are then multiple transitions where thefrequency-dependent terms can change sign and either add to or subtractfrom the baseline torque of case 1. Moreover, each torque T_(x′),T_(y′), and T_(z′) has a unique frequency where the torques must go tozero. For example, if ϕ_(x′)=ϕ_(y′)=ϕ_(z′)=0 then Eqs. 332-334 become

$\begin{matrix}{T_{x^{\prime}} = {\frac{\epsilon_{med}V}{2}E_{y^{\prime}}{E_{z^{\prime}}( {\chi_{{Ry}^{\prime}y^{\prime}} - \chi_{{Rz}^{\prime}z^{\prime}}} )}}} & (361) \\{T_{y^{\prime}} = {\frac{\epsilon_{med}V}{2}E_{z^{\prime}}{E_{x^{\prime}}( {\chi_{{Rz}^{\prime}z^{\prime}} - \chi_{{Rx}^{\prime}x^{\prime}}} )}}} & (362) \\{T_{z^{\prime}} = {\frac{\epsilon_{med}V}{2}E_{y^{\prime}}{E_{z^{\prime}}( {\chi_{{Rx}^{\prime}x^{\prime}} - \chi_{{Ry}^{\prime}y^{\prime}}} )}}} & (363)\end{matrix}$

But at equilibrium, the sum of the torques along each coordinatedirection must sum to zero. For example the torque T_(x′)=0. Therefore,assuming that E_(y′)≠0 and E_(z)≠0 then (χ_(Ry′,y′)−χ_(Rz′z′))=0. Thiscan be written as

$\begin{matrix}{\lbrack {\frac{( {M_{xx} - M_{zz}} )( {1 - \beta} )( {\epsilon_{mR} - \epsilon_{pR}} )}{{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{zz},\beta} \rbrack}{g\lbrack {{\epsilon \;}_{mR},\epsilon_{pR},M_{zz},\beta} \rbrack}} + \frac{\frac{{\epsilon_{mR}\sigma_{pR}} - {\epsilon_{pR}\sigma_{mR}}}{{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{zz},\beta} \rbrack}{g\lbrack {{\sigma \; {mR}},\sigma_{pR},M_{zz},\beta} \rbrack}}}{1 + {( \frac{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{zz},\beta} \rbrack}{g\lbrack {\sigma_{mR},\sigma_{pR},M_{zz},\beta} \rbrack} )^{2}\omega^{2}}} - \frac{\frac{{\epsilon_{mR}\sigma_{pR}} - {\epsilon_{pR}\sigma_{mR}}}{{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{xx},\beta} \rbrack}{g\lbrack {{\sigma \; {mR}},\sigma_{pR},M_{xx},\beta} \rbrack}}}{1 + {( \frac{g\lbrack {\epsilon_{mR},\epsilon_{pR},M_{xx},\beta} \rbrack}{g\lbrack {\sigma_{mR},\sigma_{pR},M_{xx},\beta} \rbrack} )^{2}\omega^{2}}}} \rbrack = 0} & (364)\end{matrix}$

which is a biquadratic equation in ω that may have as many as foursolutions. Moreover, it can be shown that the condition for a guaranteeof frequency dependent change of stability axis in the above example(ϕ_(x′)=ϕ_(y′)=ϕ_(z′)=0) is that at least one of the following relationshold

$\begin{matrix}{\frac{\epsilon_{pR}}{\epsilon_{mR}} < 1 < \frac{\sigma_{pR}}{\sigma_{mR}}} & (365) \\{\frac{\sigma_{pR}}{\sigma_{mR}} < 1 < \frac{\epsilon_{pR}}{\epsilon_{mR}}} & (366)\end{matrix}$

If neither relation holds then a change of stability axis is notguaranteed, but is still possible. For a triaxial spheroid a welldefined stability axis occurs when none of the principle torques{T_(x′), T_(y′), T_(z′)} have the same sign. For example, {T_(x′), <0,T_(y′)>0, T_(z′)>0} has a stable axis, but {T_(x′)>0, T_(y′)>0,T_(z′)>0} does not. The four real-valued solutions (if they exist) aregenerically labeled as ω=ω_(x′) here, for which T_(x′)=0. It is possibleto select a harmonic DEP frequency so that the torque is non-zero, i.e.immediately above and below ω_(x′) such that the sign of the torquereverses. Therefore, to control the direction of a particle'sorientation about the x′ axis simply apply a DEP harmonic electric fieldthat oscillates at a frequency that is above or below ω_(x′). Forexample, the torque as a function of frequency T_(x′)(ω_(x′))=0,T_(x′)(ω_(x′)+δω_(x′))>0, and T_(x′)(ω_(x′)−δω_(x′))<0. This analysiscan be done for each axis of a NP in its eigen symmetry basis. Thus,knowing the zero-torque cross-over frequencies {ω_(x′), ω_(y′), ω_(z′)}and the stability axes allows the application of electric fields atappropriate frequencies to re-orient particles in a coordinated way.

As the principle of superposition applies it is possible to use severaldifferent frequency electric-field excitations simultaneously to inducedifferent torques along each axis separately. In principle, it istherefore possible to adjust a plurality of DEP frequencies so that a NPorientation not aligned with a NP's principle axes becomes the stableaxis of the NP.

For the avoidance of doubt, a change in stability axis when there isonly one DEP frequency applied means that one of the eigen symmetrybasis axis is changed to another of the eigen stability stable axis.Said another way, for the specific case of a triaxial spheroid one ofthe principle axes of the NP of length {a, b, c} changes to another oneof the the stable axes {a→b, b→c, c→a, b→a, c→b, a→c}. This behavior isalso found in more complex NPs, such as NPs with shells, where there aremore degrees of freedom and more transition frequencies.

From the point of view of controlling light, a change in stability axismeans the effective RI changes as the optical radiation field sees a neworientation of the NP. Typically, capacitor-like transparent flat-plateelectrodes can provide a uniform electric field and then the randomlyoriented particles in a colloid between the plates will align to aspecific orientation due to a uniform DEP excitation field at a specificfrequency. Then the DEP frequency changes and the particles start toorient to a new direction in the uniform field between the plates. Thisis even true in the case where the particles are aligned with theelectric field in an unstable but zero-torque orientation as even aminor perturbation of the particle will send the particle into areorientation process. This reorientation process can continue until thenew stability axis is reached. Alternately, to stop the orientationprocess before the new stability orientation is reached (1) the DEPelectric field can be shut off, (2) the DEP electric field can bechanged to a zero-torque crossover frequency {ω_(x′), ω_(y′), ω_(z′)},(3) the NPs can be locked in place by the breaking action of using DEPtranslation to increase the compactness β (in case the volume fraction)so that dipole screening locks the orientation of each particle in placeand (4) the DEP excitation can include additional frequency componentsso that a non-principle axis of a NP becomes the stable axis.

Method (4) of the prior paragraph has the interesting property that thestable non-principle axis is reached as the steady-state solution of theFokker-Planck diffusion equation forced by a multi-frequency DEPexcitation see discussion associated with Eq. 379. This is importantbecause it allows a non-synchronous perturbation of parked particles(e.g. NP perturbation by thermal Brownian movements) in a colloid thatis energized by a field that can only come from a single unmovabledirection in the observer's frame of reference.

For example, a prolate spheroid that is between transparent capacitorplates (having the observer's frame of reference) may be parked in astable position with its long axis normal to the capacitor plates due toits prior DEP excitations. Thus, it is in a stable orientation havingE_(x′)=0 and E_(y′)=0 and only E_(z′)≠0. Then a new DEP field is applied(it has a fixed direction in the capacitor-like structure) with the hopeof placing the NP in a particular orientation. However, it takes a timedelay τ₁ on particle-1 for a suitable nudge by the Brownian movements tostart a reorientation of particle-1 with electric field perturbationsE_(x′)=δE_(x′) and E_(y′)=δE_(y′). However, it might take a time delayτ₁ for this to occur for particle-2, τ₃ for for particle-3, and so on.Thus, there is no need to be concerned about how long it takes for theinitiation of the reorientation process when a steady-state endcondition ensures that the desired orientation is achieved. This can beexploited to significantly simplify PSO optical systems in someapplications.

Intermediate Summary

In the interest of condensing the material as much as possible theconcepts of force, torque, induced dipole moment, electricsusceptibility, polarizability, depolarization, and electricsusceptibility are brought together below. From Eqs. 306, 312, 71 theequations governing the DEP process are

$\begin{matrix}{F = {\frac{\epsilon_{med}v}{2}{{Re}\lbrack {\overset{\_}{\overset{\_}{\chi}}{E_{in} \cdot {\nabla E_{in}^{*}}}} \rbrack}}} & (367) \\{T = {\frac{\epsilon_{med}v}{2}{{Re}\lbrack {\overset{\_}{\overset{\_}{\chi}}E_{in} \times E_{in}^{*}} \rbrack}}} & (368) \\{\overset{\_}{\overset{\_}{\chi}} = {( {N\overset{\_}{\overset{\_}{\alpha}}} )\lbrack {\overset{\_}{\overset{\_}{I}} - {\overset{\_}{\overset{\_}{M}}( {N\overset{\_}{\overset{\_}{\alpha}}} )}} \rbrack}^{- 1}} & (369) \\{{N\overset{\_}{\overset{\_}{\alpha}}} = {\lbrack {{{\overset{\_}{\overset{\_}{\epsilon}}}_{par}\overset{\_}{\overset{\_}{M}}} + {\epsilon_{med}( {\overset{\_}{\overset{\_}{I}} - \overset{\_}{\overset{\_}{M}}} )}} \rbrack^{- 1}( {{\overset{\_}{\overset{\_}{\epsilon}}}_{par} - {\overset{\_}{\overset{\_}{I}}\epsilon_{med}}} )}} & (370)\end{matrix}$

and from Eq. 270-273 the equations governing the optical process are

$\begin{matrix}{{\langle\overset{\_}{\overset{\_}{n}}\rangle} \approx {{n_{L}\overset{\_}{\overset{\_}{I}}} + {\lbrack {{\overset{\_}{\overset{\_}{n}}}_{P} - {n_{L}\overset{\_}{\overset{\_}{I}}}} \rbrack {\overset{\_}{\overset{\_}{\sigma}}}_{P}v_{P}}}} & (371) \\{{\overset{\_}{\overset{\_}{n}}}_{P} = {\overset{\_}{\overset{\_}{\xi}}{\overset{\_}{\overset{\_}{n}}}_{P}^{\prime}{\overset{\_}{\overset{\_}{\xi}}}^{- 1}}} & (372) \\{{\overset{\_}{\overset{\_}{\sigma}}}_{P} = {\overset{\_}{\overset{\_}{\xi}}{\overset{\_}{\overset{\_}{\sigma}}}_{P}^{\prime}{\overset{\_}{\overset{\_}{\xi}}}^{- 1}}} & (373) \\{\overset{\_}{\overset{\_}{\xi}} = {\int{\int{{\overset{\_}{\overset{\_}{W}}( {\theta,\varphi} )}{f_{T}( {r,t,\theta,\varphi,\omega_{j}} )}d\; \theta \; d\; {\varphi.}}}}} & (374) \\{{\overset{\_}{\overset{\_}{\sigma}}}_{P}^{\prime} = \begin{bmatrix}\frac{1 - M_{P,{x^{\prime}x^{\prime}}}}{1 - {M_{L,{x^{\prime}x^{\prime}}}v_{L}} - {M_{{Px}^{\prime}x^{\prime}}v_{P}}} & 0 & 0 \\0 & \frac{1 - M_{P,{y^{\prime}y^{\prime}}}}{1 - {M_{L,{y^{\prime}y^{\prime}}}v_{L}} - {M_{{Py}^{\prime}y^{\prime}}v_{P}}} & 0 \\0 & 0 & \frac{1 - M_{P,{z^{\prime}z^{\prime}}}}{1 - {M_{L,{z^{\prime}z^{\prime}}}v_{L}} - {M_{{Pz}^{\prime}z^{\prime}}v_{P}}}\end{bmatrix}} & (375)\end{matrix}$

The next sections connect the forces and torques of DEP to the opticalproperties of the colloid by means of a diffusion processes.

Forced Translational Diffusion

Up to now we have been focused on the force and torque on a singleparticle. However, in this section and in the next section, colloidsconsisting of up to tens of thousands of NPs per cubic micron areconsidered. For example, if the NPs are spherical with a diameter of 30nm then upwards of 30,000 NP may be in a cubic micron. More or lessparticles are possible depending on size, shape, composition, and theresults of a forced diffusion process.

Consider a quasi electrostatic process where E_(R) and E₁ are a functionof position r, a non-optical excitation frequency c derived fromelectrodes or other sources, and even time t for slow variations whenthe time is very much larger than (2π/ω). Thus, for a colloid we canwrite F=F(r, t, ω) and if the driving electric field has many frequencycomponents then the total force is ∫F(r, t, ω)dω and the resultingsaturation drift velocity in the viscous liquid is v(r, t)=γ∫F(r, t,ω)dω, where γ=1/(6πηb) is the intrinsic mobility by Stokes Law and r isthe dynamic viscosity. The vector particle flux for a DEP process isthen j_(dep)(r, t)=u(r, t)v(r, t), where a is the number of particlesper unit volume, i.e. particle concentration. Additionally, thethermally induced diffusion flux is j_(dif)(r, t)=−D∇u(r, t), where D isthe Stokes-Einstein diffusion coefficient. The total particle flux isthen j(r, t)=j_(dep)(r, t)+j_(dif)(r, t). The source-free continuityequation ∂_(t)u+∇·j=0 then gives

$\begin{matrix}{{\frac{\partial u}{\partial t} - {D{\nabla^{2}u}}} = {{- \nabla} \cdot {\lbrack {{\mu\gamma}{\int_{- \infty}^{+ \infty}{{F( {r,t,\omega} )}d\; \omega}}} \rbrack.}}} & (376)\end{matrix}$

This is a Fokker-Planck type integro-differential equation describingforced diffusion. Moreover, as the particle volume fraction is

v _(P)(r,t)=νu(r,t).  (377)

where ν is the volume of a nanoparticle in an ideal single-sizedmono-dispersed colloid.

Then by inserting Eq. 377 into Eq. ??, observing that v_(P)=v₀ƒ_(F),where ƒ_(F)=ƒ_(F)(r, t, ω) is the probability mass function (PDF) for aforce process, and v₀ is the average homogeneous volume fraction of theNPs in the colloid control volume then we can write the Fokker-Planckequation in terms of the PDF ƒ_(F). Additionally, in anticipation ofextending the results here to torques we can write ∇→∇_(X), to remindthe reader that this is a translational coordinate system such as (butnot limited to) cartesian coordinates

$\begin{matrix}{{\frac{\partial f_{F}}{\partial t} - {D_{F}{\nabla_{X}^{2}f_{F}}}} = {{- \nabla_{X}} \cdot \lbrack {f_{F}\gamma_{F}{\int_{- \infty}^{+ \infty}{{F( {r,t,\omega} )}d\; \omega}}} \rbrack}} & (378)\end{matrix}$

where D_(F) is the diffusion coefficient with a subscript of F toassociate it with forced diffusion and similarly for γ_(F). Thus, bycontrolling ƒ_(F)(r, t) and translational diffusion the refractive indexis also controlled. See for example Eq. 5 of the introduction.

Forced Orientational Diffusion

A similar set of considerations leads to NP rotation and orientationdiffusion in a spherical coordinate system, which is represented by∇_(Θ). Thus, the equation that governs rotational and orientationaldiffusion is given by

$\begin{matrix}{{\frac{\partial f_{F}}{\partial t} - {D_{T}{\nabla_{\Theta}^{2}f_{T}}}} = {{- \nabla_{\Theta}} \cdot \lbrack {f_{T}\gamma_{T}{\int_{- \infty}^{+ \infty}{{T( {r,t,\omega} )}d\; \omega}}} \rbrack}} & (379)\end{matrix}$

where ƒ_(T) is the PDF for torques and rotational diffusion. See forexample Eq. 6. Thus, by controlling ƒ_(T)(r, t) and orientationaldiffusion the refractive index is also controlled.

Light Scattering

This section, which focuses on how NP orientation and structure impactlight scattering, is developed intuitively. More formal scatteringtheory is avoided here due to the exceptionally long length of thetheoretical development. That said the results presented here can bederived from first principles via Maxwell's equations.

Light only scatters if there is a change in the dielectric constant(Δ∈). Moreover, this scattering might be large or small therefore therelative change in the RI is given by (Δ∈/∈_(L)) and is the quantity ofinterest with respect to scattering amplitude. Generalizing to arbitraryNP shape and orientation a tensor (Δ∈/∈)=(∈ _(P)−I∈_(L))/∈_(L) describesthe scattering amplitude, where ∈_(L) and ∈ _(P) are the liquid andsolid NP dielectric constants, and I is the identity matrix. Forexample, by using Eq. 268 and assuming that the NPs are isotropic weobtain in the eigen symmetry basis that

$\begin{matrix}{\frac{\Delta \overset{\_}{\overset{\_}{\epsilon}}}{\epsilon_{L}} = {{2( \frac{n_{p} - n_{L}}{n_{L}} )v_{P}{\overset{\_}{\overset{\_}{\sigma}}}_{P}} + {( \frac{n_{p} - n_{L}}{n_{L}} )^{2}v_{P}^{2}{\overset{\_}{\overset{\_}{\sigma}}}_{P}^{2}}}} & (381) \\{{\approx {2( \frac{n_{p} - n_{L}}{n_{L}} )v_{P}{\overset{\_}{\overset{\_}{\sigma}}}_{P}}},} & (382)\end{matrix}$

and then after conversion to the observer's basis by ∈=W ∈′W ⁻¹ we findthat

$\begin{matrix}\begin{matrix}{{\overset{\_}{\overset{\_}{\epsilon}}}^{\prime} =} & {{{\overset{\_}{\overset{\_}{n}}}^{\prime}{\overset{\_}{\overset{\_}{n}}}^{\prime}}} \\{=} & {{{n_{L}^{2}\overset{\_}{\overset{\_}{I}}} + {2{n_{L}( {n_{p} - n_{l}} )}v_{P}{\overset{\_}{\overset{\_}{\sigma}}}_{P}^{\prime}} + {( {n_{P} - n_{L}} )^{2}{v_{P}^{2}( {\overset{\_}{\overset{\_}{\sigma}}}_{P}^{\prime} )}^{2}}}}\end{matrix} & (380)\end{matrix}$

which varies as a function of space r and time t within the colloid.

Next, consider the phase of a light ray scattered from a NP. The phaseof a single-wavelength of light λ₁ from its source to the NP is (2πr₁/λ₁)=k₁r₁, where r₁ is the distance to the NP from the source.Similarly, (2π r₂/λ₂)=k₂r₂, is the distance between the NP and observer.Note λ₁ and λ₂ may be different due to Doppler shifts from movingparticles. Thus, NP translation imparts a phase change from source to NPto observer. If it is assumed that a light source and an observer areequidistant from a colloid control volume and electrically far from thescattering NP then r=r₁=r₂≈R and phase factor of Exp[−i(k₂−k₁)·r′] areintroduced by a volume element of the colloid to the light field. Wherer′ is the position of the scattering volume element. Assuming an inputplane wave of E₁=E₁{circumflex over (m)}₁e^(−i(k) ¹ ^(·r−ω) ¹ ^(t)) andan output polarization analyzer in the direction n₂. Then on applyingHuygen's principle and summing over the scattering volume elements usingexpanding spherical wavelets modified by the induced phase from thelocal motion of NPs we get

$\begin{matrix}{E_{2} = {C\underset{\underset{{{Huygen}'}s\mspace{14mu} {{Prin}.}}{}}{\frac{E_{1}e^{- {i{({{k_{2}R} - {\omega_{2}t}})}}}}{4\pi \; R}}{\int_{{vol}.}{\overset{\overset{{Phasor}\mspace{14mu} {Modification}\mspace{14mu} {of}\mspace{14mu} {Spherical}\mspace{14mu} {Wavelet}\mspace{14mu} {in}\mspace{14mu} d^{3}r^{\prime}}{}}{\underset{\underset{{Amplitude}\mspace{14mu} {by}\mspace{14mu} {NP}\mspace{14mu} {Rotation}}{}}{\lbrack {{\hat{m}}_{2} \cdot \frac{\Delta {\overset{\_}{\overset{\_}{\epsilon}}( {r^{\prime},t} )}}{\epsilon_{L}} \cdot {\hat{m}}_{1}} \rbrack}\underset{\underset{{Phase}\mspace{14mu} {by}\mspace{14mu} {NP}\mspace{14mu} {Translation}}{}}{e^{{- {i{({k_{2} - k_{1}})}}} \cdot r^{\prime}}}}d^{3}r^{\prime}}}}} & (383)\end{matrix}$

The proportionality constant C can be shown to be C=−k₂ ² by a morerigorous analysis. Therefore the full result for the scattered electricfield is rewritten as

$\begin{matrix}{E_{2} = {{- k_{2}^{2}}\frac{E_{1}e^{- {i{({{k_{2}R} - {\omega_{2}t}})}}}}{4\pi \; R}{{\hat{m}}_{2} \cdot \underset{\underset{{Fourier}\mspace{14mu} {Transform}\mspace{14mu} {of}\mspace{14mu} {tensor}\mspace{14mu} \Delta \overset{\_}{\overset{\_}{\epsilon}}}{}}{\lbrack {\int_{{vol}.}{\frac{\Delta {\overset{\_}{\overset{\_}{\epsilon}}( {r^{\prime},t} )}}{\epsilon_{L}}e^{{- {iq}} \cdot r^{\prime}}d^{3}r^{\prime}}} \rbrack} \cdot {\hat{m}}_{1}}}} & (384)\end{matrix}$

where q=k₂−k₁. Also, by way of definition let's define the spatialFourier Transform of the tensor Δ∈ and its projections onto {circumflexover (m)}₁ and {circumflex over (m)}₁ as

δ∈(q,t)=∫_(vol)Δ∈(r′,t)e ^(−iq·r′) d ³ r′  (385)

δ∈_(1,2)(q,t)={circumflex over (m)} ₂·δ∈(q,t)·{circumflex over (m)}₁  (386)

This provides the compact form for the scattered electric field as

$\begin{matrix}{{E_{2}( {q,t} )} = {{- k_{2}^{2}}\frac{E_{1}e^{- {i{({{k_{2}R} - {\omega_{2}t}})}}}}{4{\pi\epsilon}_{L}R}{{{\delta\epsilon}_{1,2}( {q,t} )}.}}} & (387)\end{matrix}$

Thus, the Fourier Transform of the spatial variation of the change inthe dielectric constant tensor projected onto the input and outputpolarizations of each differential element of the colloid map the inputelectric fields to the output electric fields. Note that the magnitudeof q=k₂−k₁ may be approximated for the case when Doppler shifts aresmall so we can write q·q=k₂ ²−2k₁·k₂+k₁ ²≈2k₁ ²(1−cos θ)=4k₁ ²sin²(θ/2) when Doppler shifts of NPs are small and k₁≈k₂. Therefore, theBragg condition for scattering from a random medium becomes

$\begin{matrix}{q = {2k_{1}\mspace{14mu} {{\sin \lbrack \frac{\theta}{2} \rbrack}.}}} & (388)\end{matrix}$

In simple words, the colloid acts as a plurality of ordered scatteringarrays that are randomly placed into the control volume.

Finally, the irradiance I at the observer's position is derived byexploiting the WienerKhinchin theorem, which states that the expectedvalue of the autocorrelation function R_(EE) of the electric field andthe power spectral density SEE are Fourier Transform pairs. Inparticular, for a wide sense stationary signal

$\begin{matrix}{{I(\omega)} = {\frac{1}{2Z}{S_{EE}(\omega)}}} & (389) \\{{S_{EE}(\omega)} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{R_{EE}(\tau)}e^{{- i}\; \omega \; t}d\; \omega}}}} & (390) \\\begin{matrix}{{R_{EE}(\tau)} = {\langle{{E^{*}(0)}{E(\tau)}}\rangle}} \\{= {\lim\limits_{tarrow\infty}{\frac{1}{T}{\int_{0}^{T}{{E^{*}(t)}{E( {t + \tau} )}{dt}}}}}} \\{= {\int_{- \infty}^{+ \infty}{{S_{EE}(\omega)}e^{{+ i}\; \omega \; t}d\; \omega}}}\end{matrix} & (391)\end{matrix}$

where Z=√{square root over (μ₀/∈_(eff))} ohms is the wave impedance.Thus, we see that there exists a scattering process, inclusive with aBragg condition, Fourier optics kernel, and Huygen's principle via Eq.387, and that is related to the linear tensor form of the RI and DEP viaEq. 382 and Eqs. 378-379.

Colloids With Memory

A particle that is translated, oriented, and deformed may have inherentmemory of its state as it will tend to stay in its final position,orientation, and shape for some time, even under the influence ofthermal Brownian movement because the viscosity of the medium may byhigh-enough to “pin” or “park” the particle into its position,orientation, and shape. This has a capacitor-like effect, where arelaxation of the state occurs over time when the system is not forcedotherwise. This property can allow time division multiplexing ofelectrodes that impress DEP fields. Thus, a pixelated display-likearrangement of electrodes may be able to refresh the state of a thinoptical device before the state of a particle has changed very much.Unlike a display that uses capacitance to hold the state of the pixel,the PSO device can use viscosity to perform very much the same function.Also as noted before it is possible to park the orientation of NPs byusing a large compactness β (via a large volume fraction) and dipolemoment shadowing, which also provides a kind of memory.

Ray Trajectories in a Graded Refractive Index

As has been discussed, NP translation, orientation, rotation, anddeformation will provide light scattering that is effectively (but notperfectly) the same as having an effective refractive index. In thissection a Graded Refractive Index (GRIN) ray tracing is developed and inlater sub-sections it is used to develop a general beam-steering andfocusing panel.

Consider light that passes through a GRIN medium and write Maxwell'sEquations Eqs. 10-16 with no light sources, whereby

$\begin{matrix}{{\nabla{\times ɛ}} = {{- \mu}\frac{\partial\mathcal{H}}{\partial t}}} & (392) \\{{{\nabla{\cdot ɛ}} = 0}{and}} & (393) \\{{\nabla{\times \mathcal{H}}} = {{+ \epsilon}\frac{\partial ɛ}{\partial t}}} & (394) \\{{\nabla{\cdot \mathcal{H}}} = 0.} & (395)\end{matrix}$

Take Eqs. 392-393 and multiply through by √{square root over (∈/2)}.Also, take Eqs. 394-395 and multiply through by i√{square root over(μ/2)}, where i=√{square root over (−1)}. Then we can add the curl anddivergence equations together respectively to obtain

$\begin{matrix}{{{\nabla{\times ( \frac{{\sqrt{\epsilon}ɛ} + {i\sqrt{\mu}}}{\sqrt{2}} )}} = {{- i}\sqrt{\epsilon\mu}\frac{\partial}{\partial t}( \frac{{\sqrt{\epsilon}ɛ} + {i\sqrt{\mu}}}{\sqrt{2}} )}}{and}} & (396) \\{{{\nabla{\cdot ( \frac{{\sqrt{\epsilon}ɛ} + {i\sqrt{\mu}}}{\sqrt{2}} )}} = 0},} & (397)\end{matrix}$

where ξ and

are real-valued functions of space and time. Let us also define thecomplex vector electromagnetic amplitude as

$\begin{matrix}{\psi = {\frac{{\sqrt{\epsilon}ɛ} + {i\sqrt{\mu}}}{\sqrt{2}}.{and}}} & (398) \\{ɛ = \frac{\psi + \psi^{*}}{\sqrt{2\epsilon}}} & (399) \\{{= \frac{\psi - \psi^{*}}{\sqrt{2\epsilon}}},} & (400)\end{matrix}$

where the asterisk indicates the complex conjugate operation. As aconsequence

$\begin{matrix}{{\nabla{\times \psi}} = {i\sqrt{\epsilon\mu}\frac{\partial\psi}{\partial t}}} & (401) \\{{\nabla{\cdot \psi}} = 0.} & (402)\end{matrix}$

However, because of the vector identity ∇·(∇×ψ)=0 there is really onlyone equation as can be seen by taking the divergence of Eq. 401,therefore

$\begin{matrix}{{{\nabla{\times \psi}} = {\frac{i}{c}\frac{\partial\psi}{\partial t}}},} & (403)\end{matrix}$

where c=1/√{square root over (∈μ)}. On rearranging and multiplyingthrough by h and using the fundamental quantum assumption Δξ=hω (whereΔξ is the change in energy in this context and is not to be confusedwith the electric field) and since k={tilde over (ω)}/c we obtain

$\begin{matrix}{{{\lbrack {\frac{\Delta ɛ}{k}{\nabla \times}} \rbrack \psi} = {i\; \hslash \frac{\partial\psi}{\partial t}}},} & (404)\end{matrix}$

which may be written in terms of an electromagnetic Hamiltonian operator

=[ΔE/k∇x], which has units of energy, so that we obtain a vector form ofSchröinger's wave equation

$\begin{matrix}{{{\psi} = {i\; \hslash \frac{\partial\psi}{\partial t}}},} & (405)\end{matrix}$

where

is Hermitian.

Consequently, Maxwell's equations are equivalent to a vector Schrodingerwave equation for wave propagation within a homogenous, isotropic andlinear material and we can identify ψ as a vector quantum probabilityamplitude for photons that also captures the polarization properties(spin angular momentum) of light. As such ψ is subject to beingconstructed by similar rules as were used to construct the originalSchrödinger equations. In particular, we can leverage the rules forcombining probability amplitudes for photons and electrons, as developedby the physicist Richard P. Feynman (e.g. see “Quantum Mechanics andIntegrals,” by Feynman and Hibbs, ISBN: 64-25171) wherein we distillthat:

-   -   Non-Interfering Paths: The probability amplitudes multiply. This        corresponds to a particle traveling along one available path by        means of subpath 1 AND subpath 2 AND subpath 3 AND etc. . . .        all stacked serially so that the value of the probability        amplitude is Exp[i (phase of prob. wave 1)]×Exp[i (phase of        prob. wave 2)]×Exp[i (phase of prob. wave 3)] . . .    -   Interfering Paths: The probability amplitudes add. This        corresponds to a particle traveling along path 1 OR along path 2        OR along path 3 OR etc. . . . all stacked in parallel. The        probability is then given by a sum of waves Exp[i (phase of        prob. wave 1)]+Exp[i (phase of prob. wave 2)]+    -   The total probability of a particle moving along a particular        path is then the magnitude square of the total probability        amplitude.

With this in mind consider light that has transitioned from a firstmedium into a colloidal medium at an optical boundary and assume thatthis light has a phase of

ϕ=k·r−{tilde over (ω)}t  (406)

where k is the vector wavenumber, r is a point in space along the pathof the light, {tilde over (ω)} is the radian frequency of the light(i.e. not the frequency of the harmonic excitation of electrodes) and tis the time. By Eq. 405 the phase of the electromagnetic wave is alsothe phase of the corresponding probability amplitude. Therefore,multiply Eq. 406 through by the reduced Planck constant h so that

Πϕ=Π·r−Π{tilde over (ω)}t  (407)

and use the fundamental quantum assumptions whereby the energytransition in an atom (or quantum dot etc. . . . ) is Δξ=Π{tilde over(ω)} and the vector momentum of the light is p=Πk. On taking the timederivative we then obtain the Legendre transformation

$\begin{matrix}{{{\hslash \frac{d\; \varphi}{dt}} = {{p \cdot v} - {\Delta ɛ}}},} & (408)\end{matrix}$

where v is the velocity. However, for the propagator of the probabilityamplitude to go from point 1 in space-time to point 2 in space-timealong an m^(th) arbitrary path requires

$\begin{matrix}{{\psi_{1,2}(m)} = {{C(m)}e^{i\frac{d\; {\varphi_{1,2}{(m)}}}{dt}\delta \; t}}} & (409)\end{matrix}$

where i=√{square root over (−1)} and C(m) is a vector of proportionalityto account for polarization effects on the m^(th) path. To go frompoint-1 to point-2 to point-3 and so on to point-N we have by the rulefor non-interfering paths that

$\begin{matrix}\begin{matrix}{{\psi_{1,N}(m)} = {{C(m)}{\psi_{1,2}(m)}{\psi_{2,3}(m)}{\psi_{3,4}(m)}\mspace{14mu} \cdots \mspace{14mu} {\psi_{{N - 1},N}(m)}}} \\{{= {{C(m)}e^{{i\Sigma}\frac{d\; {\varphi_{n,{n + 1}}{(m)}}}{dt}\delta \; t}}}} \\{ arrow{{C(m)}e^{{\frac{i}{\hslash}{\int_{t_{a}}^{t_{b}}{{({{p \cdot v} - {\Delta ɛ}})}{dt}}}}\ }} }\end{matrix} & (410)\end{matrix}$

where C(m) is the vector proportionally constant that is different foreach path and polarization. Additionally, we must also consider allpossible separate interfering paths along which the photons may alsotravel, therefore let m take on different values for different spatialpaths so that we may write a Feynman path integral for the light path asfollows

$\begin{matrix}{\mspace{76mu} {\psi = {{\psi (1)} + {\psi (2)} + {\psi (3)} + \cdots}}} & (411) \\{= {\lim\limits_{marrow\infty}{C{\int{\cdots {\int{e^{{\frac{i}{\hslash}{\int_{t_{a}}^{t_{b}}{{({{p \cdot v} - {\Delta ɛ}})}{dt}}}}\ }{dx}_{1}\ldots \; {dx}_{m}\mspace{14mu} {dy}_{1}\ldots \; {dy}_{m}\mspace{14mu} {dz}_{1}\ldots \; {dz}_{m}}}}}}}} & (412) \\{\mspace{76mu} {= {C{\int{e^{{\frac{i}{\hslash}{\int_{t_{a}}^{t_{b}}{{({{p \cdot v} - {\Delta ɛ}})}{dt}}}}\ }\; {r.}}}}}} & (413)\end{matrix}$

Note that the exponential in the above equation takes on a broad rangeof complex values as the phasor rotates in response to the evolvingwave. It can be shown that on a statistical average the phasor sum ofthe path integral is therefore zero. To avoid this and obtain anon-trivial solution we must require that the exponent is stationaryabout a specific path in space-time so that variations b of theexponential integral must obey

δ∫_(t) _(a) ^(t) ^(b) (p·v−Δξ)dt=0.  (414)

or on recognizing that p=p₀nτ and vdt=dr=τds where r is the tangentvector to the path of the ray, n is the spatial distribution of therefractive index, ds is the element of arc length and p₀=ξ₀/c is thequantum of photon momentum with a photon initially having energy ξ₀ sothat p·v di=p₀nτ·dr=p₀n{dot over (r)}·{dot over (r)}ds=p₀n ds, therefore

$\begin{matrix}{{{\delta {\int_{r_{a}}^{r_{b}}{{n(r)}{ds}}}} = {c\; \delta {\int_{t_{a}}^{t_{b}}{\frac{\Delta ɛ}{ɛ_{0}}{dt}}}}},} & (415)\end{matrix}$

which we identify as a modified form of Fermat's Principle that canaccount for the effects of quantum transitions from atoms and colloidalquantum dots. For the simple case when Δξ=0 we recover the usualFermat's Principle from Eq. 415 so that

δ∫_(r) _(a) ^(r) ^(b) n(r)√{square root over ({dot over (r)}·{dot over(r)})}ds=0  (416)

where {dot over (r)}=dr/ds and we can identify the Lagrangian as

=n(r)√{square root over ({dot over (r)}·{dot over (r)})} and thevariational problem of Eq. 415 is solved by the Euler-Lagrange equations

$\begin{matrix}{{\frac{\partial\mathcal{L}}{\partial r} - {\frac{d}{ds}( \frac{\partial\mathcal{L}}{\partial\overset{.}{r}} )}} = 0.} & (417) \\{\frac{\partial\mathcal{L}}{\partial r} = {{\sqrt{\overset{.}{r} \cdot \overset{.}{r}}\frac{\partial{n(r)}}{\partial r}} = {\frac{\partial{n(r)}}{\partial r} = {\nabla{n(r)}}}}} & (418)\end{matrix}$

because √{square root over ({dot over (r)}·{dot over (r)})}=1.Additionally,

$\begin{matrix}{\frac{\partial\mathcal{L}}{\partial\overset{.}{r}} = {{{n(r)}\frac{\partial}{\partial\overset{.}{r}}\sqrt{\overset{.}{r} \cdot \overset{.}{r}}} = {{\frac{n(r)}{\sqrt{\overset{.}{r} \cdot \overset{.}{r}}}{\overset{.}{r} \cdot \frac{\partial\overset{.}{r}}{\partial\overset{.}{r}}}} = {{{n(r)}{\overset{.}{r} \cdot I}} = {{n(r)}\overset{.}{r}}}}}} & (419)\end{matrix}$

where I is the identity matrix. We may therefore write Eq. 417 as

$\begin{matrix}{{{\frac{d}{ds}\lbrack {{n(r)}\overset{.}{r}} \rbrack} = {\nabla{n(r)}}}{or}} & (420) \\{{{{n(r)}\overset{¨}{r}} + {{\overset{.}{n}(r)}\overset{.}{r}}} = {{\nabla{n(r)}}.}} & (421)\end{matrix}$

This equation is extremely useful for deriving the trajectories of raysin a refractive index distribution. Equation 421 is now rewritten as

$\begin{matrix}{{{\overset{¨}{r} + {\lbrack \frac{\overset{.}{n}}{n(r)} \rbrack \overset{.}{r}}} = \frac{\nabla{n(r)}}{n(r)}},} & (422)\end{matrix}$

and equivalently as

$\begin{matrix}{{\overset{¨}{r} + {{\frac{d}{ds}\lbrack {\ln \mspace{14mu} {n(r)}} \rbrack}\overset{.}{r}}} = {{\nabla\lbrack {\ln \mspace{14mu} {n(r)}} \rbrack}.}} & (423)\end{matrix}$

Next, consider a situation where we specify the trajectory as a circle.While this may be quite restrictive it is a good first approximation formore elaborate ray trajectories, it is easy to parameterize as afunction of arch length, and it is practical for engineeringapplications. Therefore, take the center of the circle at a point r₀ insome arbitrary xy-plane then

$\begin{matrix}{{r(s)} = {r_{0} + {\rho {\langle{{\cos \mspace{14mu}\lbrack {2\pi \frac{s}{2\pi \rho}} \rbrack},{\sin \mspace{14mu}\lbrack {2\pi \frac{s}{2\pi \rho}} \rbrack}}\rangle}}}} & (424)\end{matrix}$

where s is the arch length and ρ is the radius of curvature of thecircle. Then we can easily calculate {dot over (r)} and {umlaut over(r)} and plug them back into Eq. 423. Next, solve the resulting twoequations in two unknowns ∂_(x)ln[n(r)] and ∂_(y) ln[n(r)] to find that

$\begin{matrix}{{\nabla\lbrack {\ln \mspace{14mu} {n(r)}} \rbrack} = {\frac{- 1}{\rho}{\langle{{\cos \mspace{14mu}\lbrack {2\pi \frac{s}{2\pi \rho}} \rbrack},{\sin \mspace{14mu}\lbrack {2\pi \frac{s}{2\pi \rho}} \rbrack}}\rangle}}} & (425)\end{matrix}$

which can be dotted into itself to yield the nonlinear partialdifferential equation

$\begin{matrix}{{( \frac{{\partial\mspace{14mu} \ln}\mspace{14mu} {n(r)}}{\partial x} )^{2} + ( \frac{{\partial\mspace{14mu} \ln}\mspace{14mu} {n(r)}}{\partial y} )^{2}} = {\frac{1}{\rho^{2}}.}} & (426)\end{matrix}$

Next, form the product solution in the usual way so that n(r)=n(x, y)=noX(x) Y(y), where n₀ is some constant RI.

$\begin{matrix}{{( \frac{d\mspace{14mu} \ln \mspace{14mu} {X(x)}}{dx} )^{2} + ( \frac{d\mspace{14mu} \ln \mspace{14mu} {Y(y)}}{dy} )^{2}} = {\frac{1}{\rho^{2}}.}} & (427)\end{matrix}$

Now, each term is only a function of one variable, and the above sum isa constant. Therefore, each term must be a constant. Therefore, set

$\begin{matrix}{\frac{d\mspace{14mu} \ln \mspace{14mu} {X(x)}}{dx} = A} & (428) \\{\frac{d\mspace{14mu} \ln \mspace{14mu} {Y(y)}}{dy} = B} & (429)\end{matrix}$

and find by simple quadrature that X(x)=e^(Ax) and Y(y)=e^(By) so thatthe full solution is

$\begin{matrix}{{n( {x,\ y} )} = {n_{0}e^{{Ax} + {By}}}} & (430) \\{{A^{2} + B^{2}} = {\frac{1}{\rho^{2}}.}} & (431)\end{matrix}$

For example, we can choose A=cos ϕ/ρ and B=sin ϕ/ρ where ϕ is the angleof the gradient relative to the x axis, so that

n(x,y)=n ₀ e ^((x cos ϕ+y sin ϕ)/ρ)  (432)

Thus, there are an infinite number of non-symmetric RI distributionsthat provide a circular arc trajectory. These trajectories areparameterized by the constants {A, B}. From an engineering point ofview, an exponentially increasing RI is not very practical as RI cannotincrease forever. However, over a limited spatial extent this can beaccomplished easily if the gradient is not too big. In fact, it is quiteperfect for DEP induced optics, where at least one of electric fieldsdecay exponentially away from an electrode array to set up a RI that hasan exponential character in one direction, and where there isconsiderable ability to vary the RI by orientation of NPs in anorthogonal direction.

Nota bene, the reader should not get the idea that a constant gradientin one direction is sufficient to provide a ray trajectory that goesaround and around in a circle. For example, Eq. 430 can always bewritten in new coordinates where the gradient is only along the new xaxis. The reader should also not get the idea that this principle of raycontrol is only limited to two dimensions. In general, we can extend thetheory developed so far into a third dimension so that

$\begin{matrix}{{( \frac{d\mspace{14mu} \ln \mspace{14mu} {X(x)}}{dx} )^{2} + ( \frac{d\mspace{14mu} l\; n\mspace{14mu} {Y(y)}}{dy} )^{2} + ( \frac{d\mspace{14mu} \ln \mspace{14mu} {Z(z)}}{dz} )^{2}} = {\frac{1}{\rho^{2}}.}} & (433)\end{matrix}$

with the solution

$\begin{matrix}{{n( {x,y,z} )} = {n_{0}e^{{Ax} + {By} + {Cz}}}} & (434) \\{{{A^{2} + B^{2} + C^{2}} = \frac{1}{\rho^{2}}},} & (435)\end{matrix}$

which describes a ray trajectory that has a radius of curvature of pover a limited circular segment and in a plane. If the direction of thegradient is given in spherical coordinates then

n(x,y,z)=n ₀ e ^((sin θcos ϕx+sin θ sin ϕy+cos θz)/ρ).  (436)

Therefore, it is possible to design different segments of an opticalsystem with different constant gradients so that light is constantlymoving along curved trajectories that are not necessarily in the sameplane. More specifically, the gradient of the RI from Eq. 430 can bepointed in any direction. For example, let B=0 then A=1/ρ and thegradient is in the x-direction. Alternately, if A=0 then B=I/ρ and thegradient is in the y-direction. Note that (1) the NP volume fraction ofv_(P) decays exponentially away from a planar electrode array andinduces a RI gradient that is exponential (just as needed) and that byusing the shape fraction tensor σ _(P)′ it is also possible to induce aRI gradient that is exponential (again, just as needed). In practicethis is very straight forward and easy to implement. Thus, usingdifferent sets of electrodes printed on planar transparent surfaces itis possible to contain a colloid in a region between transparent platesand form circular segment trajectories of the rays and steer the inputlight to any desired output direction. For the avoidance of doubt, thephrase “circular trajectory” in this paragraph means a portion of acircle, or a circular arc or segment.

The solution for circular ray trajectories provided by Eq. 436 is notunique. There are others. For example, if we were plug into Eq. 427 theX(x)=e^(−Ax) ² and Y(y)=c^(−Ay) ² then

$\begin{matrix}{{{4{A^{2}( {x^{2} + y^{2}} )}} = \frac{1}{\rho^{2}}},} & (437)\end{matrix}$

where (x²+y²)=ρ² so that A=1/(2ρ²) and

n(x,y)=n ₀ e ^((x) ² ^(+y) ² ^()/(2p) ² )  (438)

which is easily extended into three dimensions as

n(x,y,z)=n ₀ e ^((x) ² ^(+y) ² ^(z) ² ^()/(2p) ² )  (438)

Equations 436 and 439 provide a very general way to steer light.

That said, in applications where there is an input transparent medium incontact with a GRIN medium the constantly changing RI at the boundarycan send initially parallel light rays into many different directionsthat are not tangent to the desired circular trajectories in the GRINmedium. For example, imagine parallel light rays in a glass mediumhaving a RI of n₁, these light rays then impinge on a flat boundary witha GRIN medium having n(x). By Snell's law n₁ sin θ₁=n(x) sin θ(x). Ascan be seen the light just inside of the GRIN medium has an angle θ(x)that varies as a function of the position along the surface x. It willnot help to use a different direction of a gradient, e.g. Eq. 432,because all the ray trajectories in a medium with a linear exponentialgradient are circles of the same radius of curvature p, which is aconstant. What is needed is not a constant radius of curvature of thelight trajectory, but rather a variable radius of curvature that ismatched to the refracted rays at the boundary between the inputhomogeneous medium and the GRIN medium, and if needed also at the outputfrom the GRIN medium to the external environment where there is again ahomogeneous medium. This is accomplished in a subsequent sectiondescribing FIG. 8.

The One Degree of Freedom Exponential GRIN

In this section the “circular trajectory” of a RI having the form

n(r)=n ₀ c ^(Ax)  (440)

is considered again. While, in the previous sub-section the trajectorywas specified and the RI developed, in this sub-section the RI isspecified and the trajectory developed. This will give new insights intothe extent of the validity of the exponential RI distribution. Inparticular, using Eq. 440 in Eq. 423 then provides the followingequations

{dot over (u)}+Au ² =A  (441)

{dot over (v)}+Auv=0  (442)

{dot over (w)}+Auw=0  (443)

where

{dot over (x)}=u  (444)

{dot over (y)}=v  (445)

ż=w.  (446)

These equations can be integrated directly from which we obtain

$\begin{matrix}{{u(s)} = {{\tan h}( {{As} + c_{1}} )}} & (447) \\{{v(s)} = {c_{2}{{\sec h}( {{As} + c_{1}} )}}} & (448) \\{{w(s)} = {c_{3}{{\sec h}( {{As} + c_{1}} )}}} & (449) \\{{x(s)} = {c_{4} + {( \frac{1}{A} )\mspace{14mu} {\ln \mspace{14mu}\lbrack {\cosh ( {{As} + c_{1}} )} \rbrack}}}} & (450) \\{{y(s)} = {c_{5} + {( \frac{2c_{2}}{A} )\mspace{14mu} {\tan^{- 1}\lbrack {\tanh \mspace{14mu} ( \frac{{As} + c_{1}}{2} )} \rbrack}}}} & (451) \\{{z(s)} = {c_{6} + {( \frac{2c_{3}}{A} )\mspace{14mu} {\tan^{- 1}\lbrack {\tanh \mspace{14mu} ( \frac{{As} + c_{1}}{2} )} \rbrack}}}} & (452)\end{matrix}$

where {c₁, c₂, c₃, c₄, c₅, c₆} are constants of integration to bedetermined from the initial conditions of ray position and direction. Ifat s=0 the direction cosines of the propagation direction are u₀=cosθ_(u), v₀=cos θ_(v), and w₀=cos θ_(ω) then

$\begin{matrix}{c_{1} = {\tanh^{- 1}( u_{0} )}} & (453) \\{c_{2} = {v_{0}{\cosh ( c_{1} )}}} & (454) \\{c_{3} = {w_{0}{\cosh ( c_{1} )}}} & (455) \\{c_{4} = {x_{0} - {( \frac{1}{A} )\mspace{14mu} {\ln \mspace{14mu}\lbrack {\cosh ( c_{1} )} \rbrack}}}} & (456) \\{c_{5} = {y_{0} - {( \frac{2c_{2}}{A} )\mspace{14mu} {\tan^{- 1}\lbrack {\tanh \mspace{14mu} ( \frac{c_{1}}{2} )} \rbrack}}}} & (457) \\{c_{6} = {z_{0} - {( \frac{2c_{3}}{\alpha} )\mspace{14mu} {{\tan^{- 1}\lbrack {\tanh \mspace{14mu} ( \frac{c_{1}}{2} )} \rbrack}.}}}} & (458)\end{matrix}$

Note that nowhere in the above equations Eqs. 447-458 does one find thevalue for the refractive index at a coordinate point as a parameter.Only the spatial rate of change of the refractive index, i.e. through A,is included in the equations. This is an amazing property that allowsnearly parallel rays of a laser beam, which have spatial extent via abeam diameter, to continue to track almost in parallel to each otherthrough the GRIN system in the control volume.

Let's consider the special case, which may be generalized, where theinitial ray is launched from a position

x₀, y₀, z₀

direction into the z-direction so that

u₀, v₀, w₀

=

0, 0, 1

, then we have that

$\begin{matrix}{c_{1}\  = \ 0} & (459) \\{c_{2}\  = \ 0} & (460) \\{c_{3}\  = \ 1} & (461) \\{c_{4}\  = \ x_{0}} & (462) \\{c_{5}\  = \ y_{0}} & (463) \\{c_{6}\  = \ z_{0}} & (464) \\{{u(s)} = {\tanh ({As})}} & (465) \\{{v(s)}\  = \ 0} & (466) \\{{w(s)} = {{\sec h}({As})}} & (467) \\{{x(s)}\  = \ {x_{0} + {( \frac{1}{A} )\mspace{14mu} {\ln \mspace{14mu}\lbrack {\cosh ( {As} )} \rbrack}}}} & (468) \\{{y(s)}\  = \ y_{0}} & (469) \\{{z(s)}\  = \ {z_{0} + {( \frac{2}{A} )\mspace{14mu} {{\tan^{- 1}\lbrack {\tanh \mspace{14mu} ( \frac{As}{2} )} \rbrack}.}}}} & (470)\end{matrix}$

In the following analysis we shall obtain a function z=z(x) thatdescribes the ray trajectory. During these algebraic manipulations wewill find that the following identities are useful

$\begin{matrix}{{\tanh \mspace{14mu} ( \frac{A}{2} )} = \frac{\sinh \mspace{14mu} A}{{\cosh \mspace{14mu} A} + 1}} & (471) \\{{\sinh \mspace{14mu} A} = \frac{e^{A} - e^{- A}}{2}} & (472) \\{{\cosh \mspace{14mu} A} = \frac{e^{A} + e^{- A}}{2}} & (473) \\{{\cosh^{2}( \frac{A}{2} )} = \frac{{\cosh \mspace{14mu} A} + 1}{2}} & (474) \\{{\tan \mspace{14mu} A} = {\frac{1}{i}( \frac{e^{iA} - e^{{- i}A}}{e^{iA} + e^{{- i}A}} )}} & (475) \\{{\tanh \mspace{14mu} A} = \frac{e^{A} - e^{- A}}{e^{A} + e^{- A}}} & (476) \\{e^{iA} = {{\cos \mspace{14mu} A} + {i\mspace{14mu} \sin \mspace{14mu} A}}} & (477) \\{{\cos ( {2A} )} = {{2\mspace{14mu} \cos^{2}\mspace{14mu} A} - 1}} & (478) \\{{\sin ( {2A} )} = {2\mspace{14mu} \sin \mspace{14mu} A\mspace{14mu} \cos \mspace{14mu} A}} & (479)\end{matrix}$

Therefore, we can write from Eq. 468 that

cos h(As)=e ^(A(x−x) ⁰ ⁾  (480)

sin h(As)=√{square root over (e ^(2A(x−x) ⁰ ⁾)}−1  (481)

and from Eq. 470 and using Eqs. 471-473 and Eqs. 480-481 we obtain

$\begin{matrix}{{\tanh \mspace{14mu}\lbrack \frac{As}{2} \rbrack} = {{\tan \mspace{14mu}\lbrack \frac{A( {z - z_{0}} )}{2} \rbrack} = {\frac{\sinh ( {As} )}{{\cosh ( {As} )} + 1} = \frac{\sqrt{e^{2{A{({x - x_{0}})}}} - 1}}{e^{A{({x - x_{0}})}} + 1}}}} & (482)\end{matrix}$

so that

$\begin{matrix}{{e^{2{A{({x - x_{0}})}}} - 1} = {( {e^{A{({x - x_{0}})}} + 1} )^{2}{\tan^{2}\lbrack \frac{A( {z - z_{0}} )}{2} \rbrack}}} & (483)\end{matrix}$

and from Eqs. 472-473

$\begin{matrix}{{\sinh \lbrack {A( {x - x_{0}} )} \rbrack} = {2\mspace{14mu} {\cosh^{2}\lbrack \frac{A( {x - x_{0}} )}{2} \rbrack}{{\tan^{2}\lbrack \frac{A( {z - z_{0}} )}{2} \rbrack}.}}} & (484)\end{matrix}$

Now on using Eq. 474 and Eq. 471 a second time we get

$\begin{matrix}{{\tanh \lbrack \frac{A( {x - x_{0}} )}{2} \rbrack} = {{\tan^{2}\lbrack \frac{A( {z - z_{0}} )}{2} \rbrack}.}} & (485)\end{matrix}$

The next step is to use Eqs. 475-476 to obtain

1−2e ^(iA(z−z) ⁰ ⁾ e ^(−A(x−x) ⁰ ⁾ +e ^(2iA(z−z) ⁰ ⁾=0  (486)

and now expanding the terms in (z−z₀) by Euler's identity of Eq. 477,independently setting the real and imaginary parts of the resultingequation to zero to enforce the right hand side of Eq. 486 and using thetrigonometric identity of Eq. 478 or Eq. 479, and depending on if onechoses to work with the real or imaginary part of Eq. 486 respectively,we always obtain in either case a form of the equation describing theray's path through the GRIN medium as

cos [A(z−z ₀)]=e ^(−A(x−x) ⁰ ⁾,  (487)

where the electronically controllable DEP parameter A sets the radius ofcurvature of a light ray.

Equation 487 is most certainly not a circle. It is in fact a “U” shapedcurve with one side having a shape of a near perfect circle andextending with the open end of the “U” pointing in the direction of thegradient vector of the RI. To show that the trajectory is circular overa portion of the total trajectory we first take the derivative of Eq.487 with respect to x so that

$\begin{matrix}{{{\frac{d}{dx}{\cos \lbrack {A( {z - z_{0}} )} \rbrack}} = {\frac{d}{dx}e^{- {A{({x - x_{0}})}}}}}{and}} & (488) \\{\frac{dz}{dt} = {{\cot \lbrack {A( {z - z_{0}} )} \rbrack}.}} & (489)\end{matrix}$

Therefore, to make some progress on the shape of the lens to minimizelaser beam divergence for long-range accurate and precision beamsteering observe that the function

ƒ(A)=cos[A(z−z ₀)]−e ^(−A(x−x) ⁰ ⁾=0  (490)

has a Taylor series expansion about A=0 so that

$\begin{matrix}{{{A( {x - x_{0}} )} - {\frac{A^{2}}{2}\lbrack {( {x - x_{0}} )^{2} + ( {z - z_{0}} )^{2}} \rbrack} + {\frac{A^{3}}{6}( {x - x_{0}} )^{3}} + \cdots} = 0.} & (491)\end{matrix}$

Now take only the terms though second order in A as an approximation andcomplete the squares. This results in the equation

$\begin{matrix}{{( {z - z_{0}} )^{2} + ( {x - x_{0} - \frac{1}{A}} )^{2}} = \frac{1}{A^{2}}} & (492)\end{matrix}$

which is an equation of a circle having a center at (x₀+1/A, z₀) in thexz-plane and a radius of curvature ρ=1/A. Thus, it is clear that thecircular trajectory is only over part of the total ray trajectory whenthe ray is not parallel to the gradient vector. This principle is moregeneral than for just the RI distribution investigated in thissubsection and is used extensively in the description of the invention.

Embodiments

The previous section provided the detailed physics underlying ParticleSwarm Optics (PSO) in order to teach the reader basic principles ofoperation. In this section attention is placed on explanations of devicecomponents and systems. In particular, this disclosure provides a meansto control at least one of position, orientation, and deformation ofparticles in a liquid medium so that light that scatters off of theparticles can be used for practical purposes. These practical purposesinclude using scattering of light for redirecting light (i.e. focusing,defocusing, reflecting, refracting, diffracting, etc.), attenuatinglight, amplifying light, changing the spin angular momentum(polarization) and orbital angular momentum of light, and guiding lightwithin light guides and waveguides. By changing the type of NPs used thefunctional capability can change.

FIG. 6 shows three representative particles comprising a sphericalparticle 6 a, a oblate spheroidal particle 6 b, and a prolate spheroidalparticle 6 c. The particles are drawn with an identical volume and havethe same volume fraction when in isolation in a fixed volume region. Ifthe particles are solid then each particle can be translated and rotatedin a liquid medium. If the particles are soft, e.g. an emulsiveparticle, then it may also be distorted.

There are multiple ways to accomplish translation, rotation, anddeformation. That said, this disclosure focuses on dielectrophoresis(DEP), which uses non-uniform electric fields to linearly stretch andtranslate a particle; and at least one of a non-uniform electric fieldand a uniform electric field to angularly-stretch, rotate and orientatea particle. For the avoidance of doubt in this document the concept ofrotation typically means a continuous angular process and orientationtypically means a transient angular process.

The particles of FIG. 6 may be dielectric, metallic, semi-conductors,hydrocarbon polymers, silicone polymers, hybrid materials,hard-materials, soft-materials, quantum dots, plasmonic dots, viruses,multi component Janus particles, non-magnetic, magnetic, diamagentic,paramagnetic, clusters of particles that are distinct yet physicallyattached to each other, and other types and combinations of particles.Typically, these particles are electrically neutral, though this is nota requirement. One of the great advantages of using DEP is that neutralparticles can be controlled unlike electrophoresis, which requirescharged particles.

The DEP process may be used to control position and orientation of manyparticles at once. The colloid or suspension may have a mixture ofparticles of different sizes, shapes, composition, and volume fraction.For example, in FIG. 7A a perspective of a plurality of randomly placedprolate spheroids aligned along their long axis is shown. This samegroup of particles is then seen from the side in FIG. 7B. Finally, inFIG. 7C the particles are all oriented along the short axis of theparticle, which is in the direction normal to the page of the image.These images show that there can be both order and disorder in theorientation and placement of the particles. A similar image can be drawnfor oblate spheroids.

In FIG. 7B, light passing from left to right would experience a greaterrefractive index than light that is propagating normal to the page inthe same figure or in FIG. 7C. This was described in Eq. 286 andassociated text. For oblate spheroids, light that passes along thecoin's long radial direction experiences a greater RI than light thatpasses normal to the large surface of the oblate spheroid.

Next, in FIG. 8 it is demonstrated that it is possible to phase matchwaves in a homogeneous medium and a GRIN medium. In particular, a simplecase of transforming light incident at any angle directly downward isconsidered. This is not possible with conventional refractive optics. Inparticular, parallel input light rays, such as input ray 8 a, in ahomogeneous input medium 8 b having a RI of n₁ are incident on a firstboundary 8 c at an angle θ₁ to a y-axis. The input rays are propagatingin the xy-plane and strike a the first flat boundary at y=L_(y). Betweenthe first boundary 8 c and a second boundary 8 d (i.e. at y=0) there islocated a controllable colloid 8 e that can have its scatteringproperties and RI adjusted. Below the boundary at y=0 is anotherhomogeneous medium which for simplicity also has a RI of n₁. Thus, whatis described are two glass regions with parallel surfaces that sandwicha thin colloid layer in a colloid control region. This is the situationfor when glass plates with suitable electrodes sandwich a thin colloidlayer.

Input ray 8 a is refracted at a first point 8 f into the controllablecolloid 8 e. At the first point 8 f, which is on the boundary betweenthe input medium 8 b and the controllable colloid 8 e the RI of thecolloid is n(x, L_(y)), therefore by Snell's law n₁ sin θ₁=n(x, L_(y))sin θ(x, L_(y)) where θ=θ(x, L_(y)). The first refracted ray 8 g isperpendicular to the radius of curvature p, which shown as 8 h. It iseasy to see that

$\begin{matrix}{\rho = {\frac{L_{y}}{\sin \mspace{14mu} \theta} = {\frac{L_{y}\mspace{14mu} {n( {x,L_{y}} )}}{n_{1}\mspace{14mu} \sin \mspace{14mu} \theta_{1}} = \frac{L_{y}\mspace{14mu} {X(x)}{Y(y)}}{n_{1}\mspace{14mu} \sin \mspace{14mu} \theta_{1}}}}} & (493)\end{matrix}$

Let's now rewrite Eq. 427

$\begin{matrix}{{( \frac{d\mspace{14mu} \ln \mspace{14mu} {X(x)}}{dx} )^{2} + \underset{\underset{\equiv 0}{}}{( \frac{d\mspace{14mu} \ln \mspace{14mu} {Y(y)}}{dy} )^{2}}} = {\frac{1}{\rho^{2}(x)} = {( \frac{n_{1}\mspace{14mu} \sin \mspace{14mu} \theta_{1}}{L_{y}\mspace{14mu} {X(x)}{Y(y)}} )^{2}.}}} & (494)\end{matrix}$

where it has been assumed that the RI only varies in the x-direction sothat Y(y)=1—this is not necessary but it is convenient. Then

$\begin{matrix}{{{X(x)}\frac{d\mspace{14mu} \ln \mspace{14mu} {X(x)}}{dx}} = {\pm \frac{n_{1}\mspace{14mu} \sin \mspace{14mu} \theta_{1}}{L_{y}}}} & (495)\end{matrix}$

which by direct quadrature has a solution

$\begin{matrix}{{{X(x)} = {{{\pm \frac{n_{1}\mspace{14mu} \sin \mspace{14mu} \theta_{1}}{L_{y}}}x} + C}},} & (496)\end{matrix}$

where C is the constant of integration. If we define a convention thattakes the incident angle θ₁ to the left of the normal as θ₁>0 and theincident angle to the right of the normal as θ₁<0 then the solution is

$\begin{matrix}{{{n(x)} = {{{- ( \frac{n_{1}\mspace{14mu} \sin \mspace{14mu} \theta_{1}}{L_{y}} )}\mspace{14mu} x} + n_{\max}}},} & (497)\end{matrix}$

where the maximum RI occurs at x=0. Those familiar with antenna theorywill recognize that if we take this equation and multiply through by (2πL_(y)/λ₀), where λ₀ is the free space optical wavelength we find thephase lag y in radians

ψ=−(k ₁ sin θ₁)x+ψ _(max).  (498)

where λ=λ₀/n(x) and k₁ is the wavenumber in the input medium. Thus,while a RI having n(x)=n₀e^(−x/ρ) allows only one-radius-of-curvaturetrajectories, in contradistinction the RI of Eq. 497 creates an infinitenumber of different radius of curvature circles that carry the inputlight to the second boundary 8 d where the rays are all normal to theboundary and therefore easily pass to a homogeneous output medium 8 ihaving output rays, and example of which is 8 j. Also, at the dashedline x=L_(x) the lowest RI available is reached and the RI has to be setto it maximum value n_(max). Thus, the RI is varied in a specific wayalong the x-direction to create a phase screen that is capable ofsteering the light.

There are many variations on this idea. A first example, is how tochoose the RI distribution to allow steering the light into anydirection to maintain parallel rays. A second example is the RIdistribution to allow the adding of a conventional mirror at the secondboundary 8 d to allow for parallel ray steering of light into anydirection by controlled reflection. A third example is the use of curvedgaps that might be needed in some light steering applications, e.g. acar, boat, aircraft, or spacecraft body having light steeringcapabilities conformal to its curved surface. A fourth example isproviding a RI distribution for focusing. A fifth example is providing aRI distribution for arbitrary wavefront modification, e.g. for adaptiveoptics for astronomy and other applications. A sixth example is adaptivephase conjugation to allow light to always reflect back to its sourceindependent of surface orientation. A seventh example is beam steeringwith constraints on uniformity of light flux and {tilde over (e)}tendueconsiderations. This list is not exhaustive of all possibleapplications.

In summary, it is seen that different kinds of RI distributions can becreated to perform different types of optical functions. These RIdistributions may be implemented with a combination of translation,orientation, rotation, and deformation of NPs in DEP controlled colloid.How this is accomplished via a DEP NP orientation process is nowdescribed.

In FIG. 9 a slightly more general case of light-beam steering than FIG.8 is provided. In particular, an input light ray 9 a at an input angleα₁ is transformed into an output light ray 9 b at an output angle α₂.The input angle is measured from the normal direction at the inputsurface 9 c and α₁>0 when measured in the counter clockwise sense from anormal as shown in FIG. 9. Thus, we see that−π/2≤α₁≤+π/2. Similarly, theoutput angle is measured from a normal direction at the output surface 9d and α₂>0 when measured in the counter clockwise sense from the normal.Thus, we see that−π/2≤α₂≤+π/2. The input light ray 9 a is in an inputmedium 9 e of refractive index n₁. The output ray 9 b is in an outputmedium 9 f of refractive index n₂.

A first transparent plate 9 g of refractive index m₁ and a secondtransparent plate 9 h of refractive index m₂ sandwich a thin colloidsheet 9 i of refractive index n(x, y) located between 0≤y≤L_(a), whichforms a control region for DEP forces, torques, and stresses by DEPproocesses. The input light ray 9 a is refracted at the input surface 9c and subsequently refracts at the colloid input boundary 9 j at point(x_(a), L_(a)) as shown. The ray pass through the colloid sheet 9 i on agenerally curved trajectory due to a graded refractive index therein andrefracts at a colloid output boundary 9 k at point (x_(b), L_(b)) asshown; and then refracts into the second transparent plate 9 h andfinally refracts at the output surface 9 d. Light refracting at a point(x_(a), L_(a)) has a radius of curvature p as depicted by a first radiusline 9 m. Light refracting at a point (x_(b), L_(b)) also has a radiusof curvature p as depicted by a second radius line 9 n. A refractiveindex gradient arrow 9 o shows the positive direction of the gradient.The signed quantities α₁, α₂, and the refractive index gradient arrow 9o are all shown in the positive direction as taken by convention herein.If we take L_(y) as the thickness of the colloid layer then thefollowing relations hold

L _(y) =L _(a) −L _(b)=ρ sin θ(x _(a) ,L _(a))−ρ sin θ(x _(b) ,L_(b))  (499)

n ₁ sin α₁ =m ₁ sin ψ₁  (500)

m ₁ sin ψ₁ =n(x _(a) ,L _(a))sin θ(x _(a) ,L _(a))  (501)

n(x _(b) ,L _(b))sin θ(x _(b) ,L _(b))  (502)

Therefore,

$\begin{matrix}{\rho = \frac{L_{y}}{{\sin \mspace{14mu} {\theta ( {x_{a},L_{a}} )}} - {\sin \mspace{14mu} {\theta ( {x_{b},L_{b}} )}}}} & (503) \\{= {\frac{L_{y}}{\frac{n_{1}\mspace{14mu} \sin \mspace{14mu} \alpha_{1}}{n( {x_{a},L_{a}} )} - \frac{n_{2}\mspace{14mu} \sin \mspace{14mu} \alpha_{2}}{n( {x_{b},L_{b}} )}}.}} & (504)\end{matrix}$

If the refractive index n(_(a), L_(a))≈n(x_(b), L_(b))≈n(x) because thegradient in the RI is not too great then

$\begin{matrix}{\rho \approx \frac{L_{y}\mspace{14mu} {n(x)}}{{n_{1}\mspace{14mu} \sin \mspace{14mu} \alpha_{1}} - {n_{2}\mspace{14mu} \sin \mspace{14mu} \alpha_{2}}}} & (505)\end{matrix}$

and we can take n(x, y)=n(x)=X(x) and Y(y)=1 in Eq. 427 then

$\begin{matrix}{{( \frac{d\mspace{14mu} \ln \mspace{14mu} {X(x)}}{dx} )^{2} + \underset{\underset{\equiv 0}{}}{( \frac{d\mspace{14mu} \ln \mspace{14mu} {Y(y)}}{dy} )^{2}}} = {\frac{1}{\rho^{2}(x)} = {( \frac{{n_{1}\mspace{14mu} \sin \mspace{14mu} \alpha_{1}} - {n_{2}\mspace{14mu} \sin \mspace{14mu} \alpha_{2}}}{L_{y}\mspace{14mu} {X(x)}} )^{2}.}}} & (506)\end{matrix}$

and by direct quadrature we have that

$\begin{matrix}{{n(x)} = {{( \frac{{n_{2}\mspace{14mu} \sin \mspace{14mu} \alpha_{2}} - {n_{1}\mspace{14mu} \sin \mspace{14mu} \alpha_{1}}}{L_{y}} )x} + n_{0}}} & (507)\end{matrix}$

where no is the value of the RI at x=0. Correction terms to thisexpression can be developed if the RI gradient is large. Note that ifα₂=0 then we essentially recover Eq. 497. Again, the angles α₁ and α₂are positive when measured counterclockwise from the normal on the inputand output surfaces. Because RI cannot be constantly increased ordecreased there is a corresponding range 0<x≤L_(x) where NPs areorientated as needed to produce a desired RI and then reset over andover again as needed so that a large surface can be used to collect andsteer light. The RI then varies discontinuously similar to a saw-toothshape, and it is fully controllable.

Equation 507 says that parallel input ray, e.g. sunlight from the sun,can be steered into any direction desired. Not only in the plane of FIG.9, but in any direction by using suitable gradients in the RI.Furthermore, the linear RI term for light-beam steering can be combinedwith a quadratic RI for focusing, and higher order terms are useful forwavefront aberration corrections and adaptive optics applications.Therefore, a PSO system can in general provide a RI change in anydirection in the control volume. Some of the RI change may be due toorientation and deformation of NPs and some of the RI change may be dueto translation of NPs.

FIGS. 10A-B shows some of the different ways that light can be steeredby a colloid controlled by DEP in a thin control volume between twoglass plates. Electric field sources are not shown in these figures sothe reader can focus on on the possible trajectories of a single inputray. For the avoidance of doubt the output rays shown are not occurringat the same time, but are selected by specifying the colloid state byDEP process at a series of times.

FIG. 10A is for light transmission and FIG. 10B is for light reflection.In particular, FIG. 10A shows an input light ray 10 a that refracts intoa homogeneous and isotropic first transparent plate 10 b via an inputsurface 10 c to form a first ray segment 10 d, which then is refractedinto one curved ray trajectory at colloid input boundary 10 u. Ingeneral, this allows a continuous and curved ray trajectory group 10 e.The process of bending the light occurs in a colloid layer 10 f (havingthickness as small as roughly 10 μm for visible light) by pulling thelight in the direction of a left going RI gradient 10 g and a rightgoing RI gradient 10 h. If a RI gradient in the colloid is very strongit is even possible to trap some rays such as a first captured ray 10 i.Once a curved propagates to a colloid output boundary 10 j it isrefracted into a second transparent plate 10 k and is then againrefracted at the output surface 10 m. The result is a family of outputrays 10 n that can cover the full π radians of angular extent. Moreover,this can be extended to a full 4π steradians with additional RIgradients in the colloid layer 10 f, which point into or out of the pageof FIG. 10A. An example an output ray 10 o is provided and it isselected by the particular state of the colloid as induced by the DEPprocesses for translation, orientation, rotation, and deformation ofNPs.

FIG. 10B shows the same system as FIG. 10A except that a dielectricmirror 10 p has been added to the bottom of the colloid layer. A mirror(dielectric or metallic) could have been added to the bottom surface 10r as an alternative. The resulting output rays 10 s are shown as well asthe input ray 10 t. So in summary, the direction of an output ray isselected by the RI and its gradient in the colloid layer 10 q. Thus, isformed an electronically controllable mirror, where the output reflectedlight is directed by a DEP process.

In FIG. 11A-H we see how a desired gradient in refractive index (e.g.Eqs. 432 and 497) is implemented using the orientation of particles.This can provide a circular (or parabolic, hyperbolic, etc) trajectoryof input light within the colloid as was shown in FIG. 8 and providelinear, parabolic and other types of phase-lag distributions to controlbroadband light. In FIG. 11A a first transparent plate 11 a and a secondtransparent plate 11 b are substantially parallel and separated to forma gap, which is a control volume for a random spheroidal colloid i.e.either prolate or oblate, but not spherical NPs. This colloid is calledthe first colloid state 11 c. The colloid may have 10,000 NP per cubicmicron for control of visible light. This number of NPs is impossible toshow in FIG. 11A-H, therefore, the NPs are shown schematically toindicate orientation. Also, for the avoidance of doubt the scale of theNPs relative to all other parts is not to scale.

In one embodiment a plurality of electrodes is placed in or on thetransparent plates. This comprises a first electrode array 11 d and asecond electrode array 11 e. The electrode arrays may be pixelated orthey may be line electrodes, or arbitrary curves; depending on theapplication. It is desired to take the initially random state of colloidand organize its individual NPs to create a desired phase-lagdistribution via a RI distribution.

In FIG. 11B the system of FIG. 11A is subjected to a substantially highfrequency electric field 11 f. This creates torques on each randomlyorientated NP according to Eqs. 361-363. Assuming that the colloidcompactness β is not too large there will be a rotational diffusion, asdescribed in Eq. 379. In due course a steady state orientation of eachNP is reached with the long axis of each NP orientated along thedirection of the electric field as shown in FIG. 11C as a second colloidstate 11 g. This is due to the use of a very high DEP excitationfrequency, which has already described as always aligning the long axisof a NP with the electric field direction. For the avoidance of doubt,the particle is stationary and does not rotate either synchronously orasynchronously with the DEP field during alignment. The main feature ofIn FIG. 11B is the fields. To avoid clutter in this figure the images ofthe particles are suppressed. The main feature of In FIG. 11C are thealigned NPs.

Next, in FIG. 11D an asymmetric harmonic electric field 11 h isgenerated between neighboring electrodes as shown. The nonuniformportion of this electric field is substantially located below thesurface of the first transparent plate 11 a and the second transparentplate 11 b. What remains of the harmonic electric field is mostlyuniform and capable of aligning NPs by orientation only. The asymmetricelectric fields are scanned across the colloid (indicated by the arrowsthrough the electric fields) to form a uniform perturbation angle 11 iof the NPs and thus create a third colloid state 11 j as shown in FIG.11E. With this there are always at least two fields of the set {E_(z′),E_(y′), E_(z′)} in Eq. 361-363 that are not zero so that at least onetorque exists in a uniform and vertically directed electric field.Moreover, unlike a Brownian movement induced angular perturbation ofeach NP, all the NPs now have the exact same orientation and can bemoved in coordinated way from generalized DEP electric fields.

In FIG. 11F each of the electrode elements is allowed to providepixelated oscillating fields 11 k using at least one of amplitudevariations per pixel, frequency variations per pixel, and timevariations per pixel. This provides a different angular orientation ofNPs across the colloid and a fourth colloid state 11 m as shown in FIG.11G. In FIG. 11H a different region of the colloid is shown and it iseasy to see the region of the colloid where the phase abruptly changesdue to the limits in the range of the effective RI. This is a fifthcolloid state 11 n. The abrupt change in orientation is equivalent tothe line x=L_(x) in FIG. 8.

Next, FIG. 12 shows a cross section of an active colloid lens thatfocuses light that passes through its thin colloid layer based in theinduced orientation of nano-particles from dielectrophoresis. Thisactive colloid lens can be millimeters or many meters in diameterdepending on the application. For an example light source at infinityrelative to the lens the following are the input rays: a first input ray12 a, a second input ray 12 b, and a third input ray 12 c. Thecorresponding output rays are a first output ray 12 d, a second outputray 12 e, and a third output ray 12 f. The output rays converge to afirst focus region 12 g.

Between the input and the output rays are two solid transparent mediacomprising a first transparent solid 12 h and a second transparent solid12 i. Between the first transparent solid and the second transparentsolid is a liquid medium with NPs therein. This is called a transparentcolloid layer 12 j, which is contained in a gap providing a controlvolume that is formed between the first transparent solid 12 h and thesecond transparent solid 12 i. For optical applications the gap formedby the control volume is typically very thin, e.g. 1 μm to μm, thoughthis is not a necessity. The colloid in the gap may include NPs ofdifferent sizes, shapes, and composition. However, for the purpose ofthis example the NPs are at least one of prolate spheroids and oblatespheroids with upwards of 10,000 NPs per cubic micron as an order ofmagnitude estimation, where each NP has a maximum dimension of about 30nm for visible light, but can also be other values as needed. The numberof particles may be more in quantity or less in quantity as needed forthe application and in some cases can range all the way down to zero NPsper cubic micron in some portion of the system. The NPs are chosen to besmall enough so that Tyndall scattering is minimized and DEP forces,torques, and stresses are maximized to control the NPs orientationdistribution and spatial distribution. The Tyndall scattering is astrong function of the NP size and the dielectric contrast between theliquid and each NP. Therefore, it is possible to make larger NPs withsmaller RI contrast compared to the surrounding liquid so that the DEPforces, torques and stresses are larger for quicker dynamic response.

Note that FIG. 12 shows only a schematic representation of the physicalcomponents and the ray trajectories. In particular, the transparentcolloid layer 12 j shows large spheroidal NPs along the gap. Note, thefigure is physically not to scale, especially the size and number andrandom placement of NPs shown. That said, the transparent colloid layer12 j contains NPs, an example of which is nanoparticle 12 k that isdrawn to be either a prolate spheroid or an oblate spheroid. For bothoblate and prolate spheroids the direction of the long axis of the NP ischosen to create a specific RI for the input rays 12 a, 12 b, and 12 cat specific positions, i.e. as indicated by Eq. 286, to affect aconcentrating lens. Thus, the transparent colloid layer 12 j acts like aphase screen that adds specific phase delays to the optical wave. From aray perspective the phase delay for the second input ray 12 b is greaterthan the phase delay for the first input ray 12 a. In this way theeffective optical path length of the combined first input ray 12 a andthe first output ray 12 d is the same as the combined effective pathlength of the second input ray 12 b and the second output ray 12 e. Theeffective optical path length of the combined third input and outputrays is also the same. The result is that the focus of the lens is atthe first focus region 12 g. If NPs were to rotate or deform into a newconfiguration then the focus region could be changed to second focusregion 12 m via the fourth output ray 12 n, fifth output ray 12 o, andsixth output ray 12 p. Thus, through a light scattering process in thecolloid, the effective RI can be tuned and the direction of propagationof light controlled.

To achieve the dynamic focusing and steering of light the transparentcolloid layer 12 j, which acts like a phase screen to add phase lagalong the lens, must transform its effective RI along the lens. This canoccur by at least one of NP translation, orientation, and deformationusing a DEP process induced by at least one of a first electrode array12 q and a second electrode array 12 r. The electrode array is typicallyplaced in direct contact with the colloid or just past the first controlvolume wall 12 s and the second control volume wall 12 t. The electrodesmay include a plurality of discrete electrodes (as shown) or it may bejust one large electrode as might be useful for certain applications.For example, an augmented reality display may have 25 micron squarepixel elements, while a large concentrating solar array for smeltingsteel, desalination of water, or electricity generation may have pixelelements that are a square meter and many panels would cover a largearea.

FIG. 13A, which is not to scale, shows a top view of a thin opticalcolloid layer 13 a where regions of random NP orientation, for exampleregion 13 b, are interspersed with regions of ordered NP orientation toform a wave guide light. For example, input light 13 c is launched intoa region with high refractive index nanoparticles 13 d, due to the highRI long-axis of each NP being in the direction of the input lightpropagation. Also, the curved light path 13 e is along a region of NPshaving a low RI side of the NP facing the propagation direction of thelight. For example, a low RI NP is 13 f is shown on one side of the highrefractive index nanoparticles 13 d and a low RI NP 13 g is shown on theother side of the high refractive index nanoparticles 13 d. Thus, thelight follows the curved path where the RI is high and exits as outputlight 13 h. This is the same principle that a fiber optic works on, i.e.total internal reflection (TIR). Thus, DEP can be used to control a TIRprocess for guided waves, switching, and other applications.

FIG. 13B shows a side view of FIG. 13A. In particular, a firsttransparent plate 13 i and a second transparent plate 13 j provide a gapthat forms a colloid control volume 13 k where a thin colloid sheet islocated. The light exits substantially in one of three regions. A firstlight output region 13 m is located within the colloid. A second lightoutput region 13 n located within the first transparent plate 13 i. Athird light output region 13 o is located within the second transparentplate 13 j. The second and third light output regions are outside of thecolloid so that only the evanescent electromagnetic fields of the lightpenetrate into the colloid in the colloid control volume 13 k. Thus, ifthe transparent plates are used as a waveguide then there is minimumscattering with the colloid, but if the light is propagating in thecolloid then there is more scattering. Which method is used depends onthe application.

FIG. 13C again shows a side view of FIG. 13A. However, now the DEPsystem is shown comprising transparent electrodes 13 p to support timevarying voltages that induce electric fields that drive DEP processes,such as translation, orientation, and deformation of NPs. On theopposite side of the electrodes is a conducting transparent ground plane13 q. Each of the electrode pixels can support an independent timevarying voltage (with its own frequency, amplitude, and phase) thatinduces a time varying electric field within the colloid control volumeand this provides NP torques, i.e. see for example Eqs. 332-334. So a NPthat has an arbitrary orientation in the colloid can align to anyspecific stability axis based on a plurality of frequencies andamplitudes applied to each pixel.

In FIG. 13D, the top view of the electrode array 13 r is shown.Unenergized electrodes, such as 13 s, are shown as an empty square. HighRI energized electrodes, such as 13 t, are shown as squares with an “x”symbol therein. Low RI energized electrodes, such as 13 u, are shown assquares with a “+” symbol therein. Finally, it should be noted thatelectrodes are not a requirement as there is a version of DEP that iswithout electrodes and uses externally derived from non-local sourcesfields. This applies to all systems in this disclosure.

In FIG. 14 a different strategy for bending and steering light isprovided. In particular, a linear gradient in the RI will steer lightalong an approximately circular trajectory. This idea is exploited byrotating the NPs in such a way to induce a gradient from right to leftin the page of the image. The resulting input light 14 a moves on acurved light trajectory 14 b to form output light 14 c.

In FIG. 15 a canonical DEP optical device 15 a for scattering inputlight is shown, including: a plurality of distinct and separateparticles, an example of which is 15 b, at least one transparent medium15 c where particles can move freely, at least one light scatteringcontrol volume 15 d, at least one source of electromagnetic fields; suchas a uniform source of electric fields comprising a first electrode 15e, a second electrode 15 f and a first harmonic voltage source 15 g;such as an electrode array 15 h having at least one harmonic voltagesource 15 i; and such as a nonuniform source of electric fieldscomprising a curved first electrode 15 j, a curved second electrode 15 kand a harmonic voltage source 15 m; which produce uniform electricfields 15 n, decaying electric fields 15 o and non-uniform electricfields 15 p; and additionally at least one transparent solid 15 q thatforms the control volume and allows light to enter or exit the controlvolume as needed. The plurality of distinct and separate particles ismixed with the at least one transparent medium to form a substantiallytransparent non-solid medium, which is a mixture, and where theplurality of distinct and separate particles can flow in the mixturewithin the at least one light scattering control volume, where there arealso electromagnetic fields from the at least one source ofelectromagnetic fields, so that at least one of forces, torques, andstresses on each of said plurality of particles may exist by means ofdielectrophoresis, which allows at least one of the position,orientation, and shape of some portion of said plurality of distinct andseparate particles to change by forced diffusion processes so that theoptical properties of said mixture can change, so that said input lightcan interact with said mixture by at least one of traversing the mixturedirectly and traversing the mixture by evanescent fields at an opticalboundary between said mixture and a region external to said at least onelight scattering control volume; whereby continuing from FIG. 16 saidinput light 16 a is transformed into output scattered light 16 b havingdifferent properties including at least one of direction, linearmomentum, spin angular momentum, polarization, orbital angular momentum,irradiance, frequency, photon energy, angular spread, power, informationcontent, all by a graded refractive index that is synthesized in bothtime and space by the rotation, translation, and deformation of aplurality of DEP controllable NPs 16 c as needed by controlling the atleast one source of electromagnetic fields for DEP processes.

Finally, it should be realized that there are “electrodeless” DEPmethods (some references call it eDEP), which use non-metallic means todistribute DEP electric fields instead of metallic electrodes todistribute the DEP electric fields. However, it is equally easy tofabricate patterned metals, dielectrics and semiconductors that directthe DEP electric fields to where they are needed. What is important isnot the electrode materials, but rather at least one source of DEPelectric fields providing the fields where needed. How the source of DEPelectric fields manages to get the DEP fields to the control volume tocoordinate the translation, orientation, rotation, and deformation ofNPs is really just a detail. Thus, at least one of metallic electrodes,dielectric electrodes, and semiconductor electrodes that are patterningas needed and connected as needed to a source of electric fields istherefore possible and are to be considered as generalized electrodesindependent of being metallic, dielectric, semiconductor or any otherclass of materials.

SPECIFICATION END NOTES Scope of Invention

First, this disclosure extends the techniques of this author's priorpatent application U.S. 2019/0353975, published on 2019 Nov. 21 andentitled,” Agile Light Control by Means of Noise, Impulse, and HarmonicSignal Induced Dielectrophoresis Plus Other Phoretic Forces to ControlOptical Shock Waves, Scattering, and the Refractive Index of Colloids;Applications Include: Solar Electricity, Solar Smelting, SolarDesalination, Augmented-Reality, LiDAR, 3D-Printing, High-Power FiberLasers, Electronic Lenses, Light Beam Steering, Robotic Vision, SensorDrones, Diffraction-Minimizing Light Beams, Power Beaming, andSoftware-Configurable Optics.” In particular, the above stated priordisclosure was for translation of particles and this disclosure is aboutextending this to particle orientation, rotation, deformation. Thisdisclosure is also about joint processes such as translation andorientation, translation and deformation of particles.

Second, while the above descriptions in each of the sections containsmany specific details for using generalized dielectrophoresis for lightcontrol, these details should not be construed as limiting the scope ofthe invention, but merely providing illustrations of some of thepossible methods, physical embodiments and applications. In particular,the present invention is thus not limited to the above theoreticalmodeling and physical embodiments, but can be changed or modified invarious ways on the basis of the general principles of the invention.

Third, every effort was made to provide accurate analysis of the physicsas part of teaching the disclosure. Nonetheless, typographical and othererrors in equations sometimes make it through reviews. The main featuresof the invention are described, however not all aspects of theunderlying physics are included due to pragmatic constraints of time andpage count. Therefore, derivations, individual equations, texturaldescriptions, and figures should be taken together so that clarity ofmeaning is ascertained from a body of information even in the event ofan unintended error in an equation or other form. Also note that many ofthe figures in the disclosure are not to scale, but are instead providedto maximize understanding of the underlying concepts. So again thetotality of the disclosure is important to consider.

Fourth, the theoretical discussion provided in this disclosure reusedsome mathematical symbols to mean different things in differentlocations of the text for historical and pragmatic reasons. The meaningis readily discernible by those skilled in the art when taken in contextof the associated descriptions.

Fifth, many potential end-use applications follow from a few physicalprinciples and a few generic embodiments. There are more potentialspecific applications than can reasonably be discussed and shown indetail with figures.

Sixth, the scope of the invention should in general be determined by theappended claims and their equivalents jointly with the examples,embodiments, and theoretical analysis provided.

Seventh, it should be obvious to those of ordinary skill that while thefocus herein is on optics controlled via Dielectrophoresis, this focusmay be extended to magnetophoresis, acoustophoresis, chemophoresis,electrophoresis, and other similar modes of particle control.

INDUSTRIAL APPLICABILITY

This invention has applicability for controlling light by controllingscattering from small particles. The control includes optical operationsas beam steering and focusing as well as general wavefront modificationwithout significant restrictions due to polarization, direction of inputlight and direction of output light. Specific applications include, butare not limited to: Light Detection and Ranging (LiDAR), electronicallyfocused camera lenses, robotic visions systems, adaptive automotiveheadlights, light-art, free-space photonic network configurations forcomputing, laser machining, laser power beaming to remotely powerdrones, beamed energy-distribution networks, 3D-printing, topographicmapping, automated inspection, dynamic holograms, remote sensing,point-to-point communications, computer displays, augmented realitydisplays, virtual reality displays, mixed reality displays, electronicpaper, sensor drones, surveying, drought monitoring sensors, aircraftcollision avoidance, 5G Light Fidelity (LiFi) networks, drone basedstructural inspection, very large aperture adjustable-membrane-opticsfor satellites & astronomy, construction site monitoring, security,laser scanning for bar code readers, optical reflectance switches fortelecommunications, solar concentrators, solar power plants, solardesalination plants, solar smelting plants, solar mining, solarradiation control windows, light-beam power combiner, laser systems,laser gyroscopes, laser machining, a manufacturing machine for makinggraded refractive index devices, software reconfigurable optics,Concentrating Solar Thermal (CST) for intense heat for industrialelectricity, steel smelting, desalination, glass, etc. without fossilfuels.

Historical Origin

The historical origin of PSO is Concentrated Solar Power (electricity)and Concentrated Solar Thermal (heat), both being developed by theinventor.

Acknowledgments

The inventor would like to express his sincere thanks and gratitude forthe support and extreme patience of his wife Susan and daughter Anora.

Reference Signs List 6a Spherical Particle 6b Oblate Spheroidal Particle6c Prolate Spheroidal Particle 8a Input Ray 8b Input Medium 8c FirstBoundary 8d Second Boundary 8e Controllable Colloid 8f First Point 8gFirst Refracted Ray 8h Radius Of Curvature 8i Output Medium 8j OutputRay 9a Input Light Ray 9b Output Light Ray 9c Input Surface 9d OutputSurface 9e Input Medium 9f Input Medium 9g First Transparent Plate 9hSecond Transparent Plate 9i Colloid Sheet 9j Colloid Input Boundary 9kColloid Input Boundary 9m First Radius Line 9n Second Radius Line 9oRefractive Index Gradient Arrow 10a Input Light Ray 10b FirstTransparent Plate 10c Input Surface 10d First Ray Segment 10e Curved RayTrajectory Group 10f Colloid Layer 10g Left Going RI Gradient 10h RightGoing RI Gradient 10i First Captured Ray 10j Colloid Output Boundary 10kSecond Transparent Plate 10m Output Surface 10n Family of Output Rays10o Example Output Ray 10p Dielectric Mirror 10q Colloid Layer 10rBottom Surface 10s Output Rays 10t Input Ray 10u Colloid Input Boundary11a First Transparent Plate 11b Second Transparent Plate 11c FirstColloid State 11d First Electrode Array 11e Second Electrode Array 11fHigh Frequency Electric Field 11g Second Colloid State 11h AsymmetricElectric Field 11i Second Colloid State 11j Third Colloid State 11kPixelated Oscillating Fields 11m Fourth Colloid State 11n Fifth ColloidState 12a First Input Ray 12b Second Input Ray 12c Third Input Ray 12dFirst Output Ray 12e Second Output Ray 12f Third Output Ray 12g FirstFocus Region 12h First Transparent Solid 12i Second Transparent Solid12j Transparent Colloid Layer 12k Nanoparticle 12m Second Focus Region12n Fourth Output Ray 12o Fifth Output Ray 12p Sixth Output Ray 12qFirst Electrode Array 12r Second Electrode Array 12s First ControlVolume Wall 12t Second Control Volume Wall 13a Top View of Colloid Layer13b Random NP Region 13c Input Light 13d High RI Nanoparticle 13e CurvedLight Path 13f Low RI NP 13g Low RI NP 13h Output Light 13i FirstTransparent Plate 13j Second Transparent Plate 13k Colloid ControlVolume 13m First Light Output Region 13n Second Light Output Region 13oSecond Light Output Region 13p Transparent Electrodes 13q TransparentGround Plane 13r Top View of Electrode Array 13s Unenergized Electrode13t High RI Energized Electrode 13u High RI Energized Electrode 14aInput Light 14b Curved Light Trajectory 14c Output Light 15a CanonicalDEP Optical Device 15b Particles 15c Medium 15d Light Scattering ControlVolume 15e First Electrode 15f Second Electrode 15g First HarmonicVoltage Source 15h Electrode Array 15i At Least One Harmonic VoltageSource 15j Curved First Electrode 15k Curved Second Electrode 15mHarmonic Voltage Source 15n Uniform Electric Fields 15o DecayingElectric Fields 15p Non-uniform Electric Fields 15q Transparent solid16a Input Light 16b Output Scattered Light 16c Controllable Nanoparticle

1. A dielectrophoresis-based device for scattering input light,comprising: (a) a plurality of distinct and separate particles; (b) atleast one transparent medium; (c) at least one light scattering controlvolume; (d) at least one source of electromagnetic fields; wherein saidplurality of distinct and separate particles is mixed with said at leastone transparent medium to form a substantially transparent non-solidmedium, which is a mixture, and where said plurality of distinct andseparate particles can flow in said mixture within said at least onelight scattering control volume, where there are also electromagneticfields from said at least one source of electromagnetic fields, so thatat least one of forces, torques, and stresses on each of said pluralityof particles may exist by means of dielectrophoresis, which allows atleast one of the position, orientation, and shape of some portion ofsaid plurality of distinct and separate particles to change by forceddiffusion processes so that optical properties of said mixture canchange, so that said input light interacts with said mixture by at leastone of traversing the mixture directly and traversing the mixture byevanescent fields at an optical boundary between said mixture and aregion external to said at least one light scattering control volume;wherein said input light can be transformed into output light havingdifferent properties including at least one of direction, linearmomentum, spin angular momentum, polarization, orbital angular momentum,irradiance, frequency, photon energy, angular spread, power, andinformation content, all by a graded refractive index that is created inboth time and space as needed by controlling said at least one source ofelectromagnetic fields.
 2. The device of claim 1, wherein said particleforces, orientations, and stress vary with frequency.
 3. The device ofclaim 1, wherein said particle orientation can be arbitrarily selectedby one or more applied frequencies of said electric fields producingdielectrophoresis.
 4. The device of claim 1, wherein said plurality ofparticles are smaller than the wavelength of light being controlled. 5.The device of claim 1, wherein said plurality of particles comprisenanostructured particles with designed shapes and scattering properties.6. The device of claim 1, wherein said plurality of particles compriseat least one of a dielectric, a metal, a semiconductor, and soft mattersuch as an emulsion.
 7. The device of claim 1, wherein said plurality ofparticles comprise optical scatterers of at least one of a quantum dot,a plasmonic dot, a photonic crystal dot, a virus, and a biologic cell,and atomically doped materials, all of which have complex internalstructures.
 8. The device of claim 1, wherein said plurality ofparticles includes particles having a distribution of different sizes,shapes, dielectric constants, refractive indices and conductivities. 9.The device of claim 1, wherein said plurality of particles comprises atleast one of optically transparent, optically opaque and opticallyreflecting particles.
 10. The device of claim 1, wherein said pluralityof particles comprises at least one of glass, diamond, titanium dioxide,corundum, barium titanate, metals, and metallic alloys.
 11. The deviceof claim 1, wherein said plurality of particles is subject to at leastone of positive material dispersion and negative material dispersion.12. The device of claim 1, wherein said plurality of particles comprise,doped transparent gain medium with atoms of at least one of dysprosium,erbium, holmium, neodymium, praseodymium, thulium, and ytterbium. 13.The device of claim 1, wherein said plurality of particles comprisephotochromatic particles.
 14. The device of claim 1, wherein saidplurality of particles comprise electret type particles.
 15. The deviceof claim 1, wherein said plurality of particles comprises at least oneof solid particles, and particles with voids therein.
 16. The device ofclaim 1, wherein said plurality of particles comprises at least one ofcage molecules, cage molecule derivatives formed by chemical bonding,and endohedral forms of cage molecules.
 17. The device of claim 1,wherein said plurality of particles comprise in part surfactants andother chemicals to help provide a stable colloid that remains separatedand does not precipitate and flocculate.
 18. The device of claim 1,wherein said plurality of particles is optically transparent over aselect spectral range.
 19. The device of claim 1, wherein said pluralityof particles comprise Janus type particles, which comprise differentmaterial properties on at least two portions of each particle, so that aself-phoretic process may self-propel and self-rotate said plurality ofparticles using energy derived from the environment in order to provideat least one of a modified distribution of particles and a modificationof the diffusion coefficient.
 20. The device of claim 1, wherein said atleast one source of electromagnetic fields provides fields that have atleast one of a harmonic signal, a plurality of harmonic signals,frequency combs, and noise.
 21. The device of claim 1, wherein said atleast one light scattering control volume has at least one opticallytransparent window to allow light to enter, exit, and interact jointlywith said mixture.
 22. The device of claim 1, wherein said at least onesource of electromagnetic fields is distributed by at least one ofmetallic electrodes, dielectric electrodes, and semi-conductorelectrodes.
 23. The device of claim 1, wherein said a plurality ofdistinct and separate particles resides in at least one of a liquid, agas, and vacuum as a continuous medium.